Non-commutative Clark measures for the free and abelian Toeplitz algebras
aa r X i v : . [ m a t h . OA ] M a r NON-COMMUTATIVE CLARK MEASURES FOR THE FREE ANDABELIAN TOEPLITZ ALGEBRAS
M.T. JURY AND R.T.W. MARTIN
Abstract.
We construct a non-commutative Aleksandrov-Clark measure for any elementin the operator-valued free Schur class, the closed unit ball of the free Toeplitz algebraof vector-valued full Fock space over C d . Here, the free (analytic) Toeplitz algebra isthe unital weak operator topology (WOT)-closed algebra generated by the componentoperators of the free shift, the row isometry of left creation operators. This defines abijection between the free operator-valued Schur class and completely positive maps (non-commutative AC measures) on the operator system of the free disk algebra, the norm-closed algebra generated by the free shift.Identifying Drury-Arveson space with symmetric Fock space, we determine the rela-tionship between the non-commutative AC measures for elements of the operator-valuedcommutative Schur class (the closed unit ball of the WOT-closed Toeplitz algebra gen-erated by the Arveson shift) and the AC measures of their free liftings to the free Schurclass. Introduction
In the classical, single-variable theory of Hardy spaces of analytic functions in the complexunit disk, D , there are natural bijections between the three classes of objects:(1) the Schur class , S , of contractive analytic functions on the complex unit disk, D ,(2) the Herglotz class , S + , of analytic functions with non-negative real part on thedisk, and,(3) the cone of positive finite Borel measures on the unit circle, T .The bijection between Schur functions, b , and Herglotz functions, H , is given by: b b − b ∈ S + ; and H H − H + 1 ∈ S , these maps are compositional inverses (we assume here that b is not the constant function b ≡ b ( z )1 − b ( z ) =: H b ( z ) = Z T zζ ∗ − zζ ∗ µ b ( dζ ) + i Im ( H b (0)) , (this is really a bijection modulo imaginary constants). In the above ζ ∗ := ζ denotescomplex conjugate. The unique positive Borel measure µ b corresponding to b , is calledthe Herglotz or Aleksandrov-Clark measure of b . More generally, for any α ∈ T , theHerglotz measure µ α := µ bα ∗ is called an Aleksandrov-Clark (AC) measure for b . Thetheory of Aleksandrov-Clark measures has played an important role in the developmentof Hardy space theory and model theory for contractions on Hilbert space, as well as incharacterizations of the Schur class [1, 2, 3, 4, 5].Given any AC measure, µ b , it is natural to consider the associated measure space L ( µ b ) := L ( µ b , T ) of measurable functions on the circle which are square-integrable withrespect to µ b , as well as the analytic subspaces H ( µ b ) , H ( µ b ) ⊆ L ( µ b ), H ( µ b ) := _ n ≥ ζ n ⊇ _ n ≥ ζ n =: H ( µ b ) , the closed linear spans of the analytic polynomials and non-constant analytic monomials,respectively. A function-theoretic argument combined with the classical distance formulaof Szeg¨o-Kolmomogoroff-Kreˇın for the distance from H ( µ b ) to the constant function 1 in L ( µ b ) shows that H ( µ b ) = H ( µ b ) = L ( µ b ) if and only if b is an extreme point of theSchur class [6, Chapter 4, Chapter 9].On the other hand, given any contractive analytic function, b , on the open unit disk, itis also natural to consider the sesqui-analytic positive kernel function k b : D × D → C : k b ( z, w ) := 1 − b ( z ) b ( w ) ∗ − zw ∗ ; z, w ∈ B d , the deBranges-Rovnyak kernel of b . Elementary reproducing kernel Hilbert space (RKHS)theory implies that there is a unique RKHS of analytic functions in the disk, H ( k b ), corre-sponding to k b , and that H ( k b ) is contractively contained in the Hardy space H ( D ). Thisspace is called the deBranges-Rovnyak space of b and we will use the standard notation H ( b ) := H ( k b ). One can also show that, in this single-variable setting, any deBranges-Rovnyak space is invariant for S ∗ , the backward shift on H ( D ) which acts as the differencequotient: ( S ∗ h )( z ) = h ( z ) − h (0) z . Here the shift, S , is the isometry of multiplication by z on H ( D ), and this operator iscentral to the study of function theory and operator theory on Hardy space [7, 8, 6].In the seminal paper [5], D.N. Clark established the following results for the case of inner b (the general versions for all Schur class functions can be found in [9, Chapter III]): Letˆ Z b denote the unitary operator of multiplication by the independent variable in L ( µ b ).The analytic subspace H ( µ b ) is invariant for ˆ Z b , and we set Z b := ˆ Z b | H ( µ b ) , an isometrywhich equals ˆ Z b if and only if b is an extreme point of the Schur class. REE ALEKSANDROV-CLARK THEORY 3
Lemma 1.1. (weighted Cauchy transform) For any contractive analytic b ∈ S , and any α ∈ T , the weighted Cauchy transform F α : H ( µ bα ∗ ) → H ( bα ∗ ) = H ( b ) defined by ( F α f )( z ) := (1 − b ( z ) α ∗ ) Z T f ( ζ )1 − zζ ∗ µ bα ∗ ( dζ ) , is a linear isometry of the analytic subspace H ( µ bα ∗ ) onto the deBranges-Rovnyak space H ( b ) . For simplicity assume b (0) = 0 and let X ∗ := S ∗ | H ( b ) . For any α ∈ T , let F α := F bα ∗ and Z α := Z bα ∗ . Theorem 1.2. (Clark’s unitary perturbations) Let b ∈ S be a contractive analytic functionin the disk (assume b (0) = 0 ). Given any α ∈ T , the weighted Cauchy transform F α intertwines the co-isometry Z ∗ α with a rank-one perturbation of X ∗ : X ∗ α := F α Z ∗ α F ∗ α = X ∗ + h· , i S ∗ bα ∗ . The point evaluation vector at , k b ≡ ∈ H ( b ) is cyclic for each X α .If b is an extreme point of the Schur class then Z α = ˆ Z bα ∗ is unitary so that each X α is arank-one unitary perturbation of the restricted backward shift X . In this case if P α denotesthe projection-valued measure of X α then µ α (Ω) = h P α (Ω)1 , i . Remark 1.3.
In the case where b is an extreme point (so that H ( µ b ) = L ( µ b )), theinverse of the weighted Cauchy transform F α implements a spectral realization for theunitary operator X α .Recently, the concept of Aleksandrov-Clark measure and all of the above results havebeen generalized to the several-variable setting of Drury-Arveson space [10] (see [11] for thevector-valued version). Here, the Drury-Arveson space, H d , consists of analytic functionson the open unit ball of d -dimensional complex space, and is a canonical several-variablegeneralization of the classical Hardy space H ( D ). We will briefly recall the relevant defini-tions in the upcoming subsection. The appropriate several-variable analogue of the Schurclass is the closed unit ball of the several-variable (analytic) Toeplitz or Hardy algebra , H ∞ d ,the (commutative) WOT-closed operator algebra generated by the Arveson d − shift on H d .(Here, note that the classical Schur class of the disk can be identified with the closed unitball of the Banach algebra H ∞ ( D ) = H ∞ of bounded analytic functions in the open disk,and that H ∞ ( D ) can be identified with the unital WOT-closed operator algebra generatedby the shift.) The Aleksandrov-Clark measures are necessarily promoted to positive linearfunctionals (or completely positive maps in the vector-valued setting) acting on a certain‘symmetrized’ operator subsystem S + S ∗ , S := S d , of A + A ∗ , where A := A d is the leftfree disk algebra, the unital norm-closed (non-commutative) operator algebra generated by M.T. JURY AND R.T.W. MARTIN the left creation operators on the full Fock space over C d . The measure space H ( µ b ) inthe several-variable setting is naturally generalized to a Gelfand-Naimark-Segal-type spacewith inner product constructed using the non-commutative AC measure, µ b , of b (as in theproof of Stinespring’s dilation theorem from [12]). With this dictionary, the classical corre-spondence between the Schur class, Herglotz functions and AC measures can be extendedto define bijections between [10, 11]:(1) The (operator-valued, several-variable) Schur class, S d ( H ) := [ H ∞ d ⊗ L( H )] ,(2) The (operator-valued, several-variable) Herglotz-Schur class, S + d ( H ), consisting ofHerglotz-Schur functions H b ( z ) := ( I − b ( z )) − ( I + b ( z ), on B d , for b ∈ S d ( H ), and,(3) The positive cone CP ( S ; H ) of all completely positive (CP) operator-valued maps µ from the symmetrized operator system S + S ∗ into L( H ).As before, if b ∈ S d ( H ), the corresponding CP map µ b ∈ CP ( S ; H ) is called the Aleksandrov-Clark (AC) map, or non-commutative AC measure, of b . These AC maps are direct several-variable generalizations of the classical AC measures.In this paper our goal is two-fold. Our first aim is to further extend the notion of a non-commutative Aleksandrov-Clark measure, the above bijection between the Schur class andAC measures, Clark’s unitary perturbations and several related results to the setting of the(left and right) free Schur class of the (left and right) free analytic Toeplitz algebra . Here, theleft (right) non-commutative or free analytic Toeplitz algebra , or more simply free Toeplitzalgebra , L ∞ d ( R ∞ d ), is the unital WOT-closed algebra generated by the left (right) creationoperators on the full Fock space, F d , over C d . As in the abelian case, we will often omitthe term analytic and call L ∞ d the left free Toeplitz algebra. The left and right (operator-valued) free Schur classes, L d ( H ), R d ( H ) are then the closed unit balls of the left and rightfree (operator-valued) Toeplitz algebras associated to vector-valued Fock space F d ⊗ H .The connection with the commutative theory is that Drury-Arveson space, H d , can benaturally identified with symmetric Fock space, H d ⊂ F d , and under this identification H d is co-invariant and full ( i.e. cyclic) for both the left and right free shifts (the row isometriesof left and right creation operators). That is, if L denotes the left free shift, L is the minimalrow isometric dilation of its compression to H d , and this compression is the commutativeArveson d − shift, S , on H d . The commutative several-variable Toeplitz algebra, H ∞ d , canthen be identified with the quotient of either the left or right free Toeplitz algebra bythe two-sided commutator ideal. Equivalently, H ∞ d can be obtained as the compression of L ∞ d or R ∞ d to symmetric Fock space H d , and this compression is a completely contractiveunital epimorphism [13]. By commutant lifting, given any commutative Schur class element b ∈ S d ( H ), there are both left and right free lifts , B L ∈ L d ( H ), B R ∈ R d ( H ) so that theirimage under the quotient map is b [14, 15]. That is, if, for example, M LB denotes leftmultiplication by B L on F d , then ( M LB ) ∗ | H d = M ∗ b , and B L , B R , b have the same norm. Of REE ALEKSANDROV-CLARK THEORY 5 course these free lifts need not be unique. We will see that left and right free lifts comein pairs B := ( B L , B R ) which are conjugate via transposition, the canonical involutionbetween the left and right free Toeplitz algebras, and that each pair, B , corresponds to aunique non-commutative Aleksandrov-Clark measure. This non-commutative AC measureis a completely positive (CP) map, µ B : A + A ∗ → L( H ).Our second goal, then, is to relate any non-commutative AC completely positive measure µ B of a transpose-conjugate pair of free lifts B = ( B L , B R ) ∈ L d ( H ) × R d ( H ) of a givencommutative b ∈ S d ( H ) with the AC map µ b acting on the symmetrized subsystem S + S ∗ ⊆ A + A ∗ as constructed in [10, 11]. In particular, we will show that any such µ B is a completelypositive extension of µ b , and that b has a unique pair of free lifts if and only if µ b , µ B are quasi-extreme in the sense of [10, 11], a property which reduces to the classical Szeg¨oapproximation property: H ( µ b ) = H ( µ b ) in the single-variable, scalar-valued setting(and which is equivalent to being an extreme point in this case) [16, 6]. This bijectivecharacterization of the set of all free lifts of a given Schur class b ∈ S d ( H ) provides analternative to the canonical deBranges-Rovnyak colligation and transfer function realizationof the commutative and free Schur classes of [17, 18, 19]. In particular our characterizationhas the advantage of providing a bijective parametrization of the set of all (generally non-unique in the commutative case) canonical deBranges-Rovnyak colligations in terms ofcertain completely positive extensions of the AC map µ b to the full free disk operatorsystem (equivalently in terms of certain free lifts of b ). In Section 7.8 we work out theprecise relationship between the canonical deBranges-Rovnyak colligations in the free andcommutative settings.1.4. Preliminaries.
Recall that Drury-Arveson space, H d , is the unique RKHS on B d :=( C d ) corresponding to the several-variable sesqui-analytic Szeg¨o kernel: k ( z, w ) := 11 − zw ∗ ; z, w ∈ B d , where in the above z = ( z , ..., z d ); w ∗ := ( w , ..., w d ), and zw ∗ := ( w, z ) C d (all innerproducts are assumed conjugate linear in the first argument).Given any RKHS H ( K ) of H -valued functions on a set X , a natural construction to con-sider is the multiplier algebra , Mult( H ( K )), of H ( K ). This is the algebra of all functions m : X → L( H ) so that mf ∈ H ( K ) for all f ∈ H ( K ). That is, the multiplier algebra isthe algebra of all functions, or multipliers , which multiply H ( K ) into itself. This algebra isclearly unital, and standard functional analytic arguments show that any multiplier, F , de-fines a bounded linear multiplication operator, M F , on H ( K ) and under this identification,Mult( H ( K )) is closed in the weak operator topology (WOT) of L( H ( K )). M.T. JURY AND R.T.W. MARTIN
The multiplier algebra, Mult( H d ), of the RKHS H d is H ∞ d , the several-variable (analytic)Toeplitz or Hardy algebra, the WOT-closure of the unital operator algebra generated bythe Arveson d -shift. Here recall the Arveson shift, S : H d ⊗ C d → H d , is the (commutative)row partial isometry S = ( S , ..., S d ) whose component operators act as multiplication bythe independent variables:( S j h )( z ) = z j h ( z ) = z j h ( z , ..., z d ); 1 ≤ j ≤ d, h ∈ H d . The several-variable
Schur class , S d = S d ( C ), is the closed unit ball of this multiplieralgebra. More generally the operator-valued Schur classes are the closed unit balls of themultipliers between vector-valued Drury-Arveson spaces: S d ( H , H ) := [Mult( H d ⊗ H , H d ⊗ H )] . We will focus on the ‘square’ case where H = H = H : S d ( H ) := S d ( H , H ) = [ H ∞ d ⊗ L( H )] , our results can be easily extended to the general rectangular setting (the rectangular Schurclasses can be embedded in square Schur classes by adding rows or columns of zeroes).Given any b ∈ S d ( H ) (or more generally any b ∈ S d ( H , H ) = [ H ∞ d ⊗ L( H , H )] ),one can construct the positive deBranges-Rovnyak kernel , k b ( z, w ) := I − b ( z ) b ( w ) ∗ − zw ∗ ; z, w ∈ B d , and the associated deBranges-Rovnyak RKHS , H ( b ). By standard RKHS theory, thesespaces are always contractively contained in H d ⊗ H .It will often be convenient to view H d as symmetric or bosonic Fock space over C d [20,Section 4.5]: First recall that the full Fock space over C d , F d , is the direct sum of all tensorpowers of C d : F d := C ⊕ (cid:16) C d ⊗ C d (cid:17) ⊕ (cid:16) C d ⊗ C d ⊗ C d (cid:17) ⊕ · · · = ∞ M k =0 (cid:16) C d (cid:17) k ·⊗ . Fix an orthonormal basis e , ..., e d of C d . The left creation operators L , ..., L d are theoperators which act as tensoring on the left by these basis vectors: L k f := e k ⊗ f ; f ∈ F d , and similarly the right creation operators R k ; 1 ≤ k ≤ d are defined by tensoring on theright R k f := f ⊗ e k . REE ALEKSANDROV-CLARK THEORY 7
The left and right free shifts are the row operators L := ( L , ..., L d ) and R := ( R , ..., R d )which map F d ⊗ C d into F d . Both L, R are in fact row isometries: L ∗ L = I F ⊗ I d = R ∗ R .The orthogonal complement of the range of L or R is the vacuum vector 1 which spans thethe subspace C =: ( C d ) ·⊗ ⊂ F d . A canonical orthonormal basis for F d is then { e α } α ∈ F d where e α = L α R α F d is the free unital semigroup or monoid on d letters.Recall here that the free semigroup, F d , on d ∈ N letters, is the multiplicative semigroupof all finite products or words in the d letters { , ..., d } . That is, given words α := i ...i n , β := j ...j m , i k , j l ∈ { , ..., d } ; 1 ≤ k ≤ n, ≤ l ≤ m , their product αβ is defined byconcatenation: αβ = i ...i n j ...j m , and the unit is the empty word, ∅ , containing no letters. Given α = i · · · i n , we use thestandard notation | α | = n for the length of the word α .For any permutation σ on n letters, one can define a unitary operator U σ on ( C d ) ⊗ C n ⊂ F d by U σ ( u ⊗ u ⊗ · · · ⊗ u n ) := u σ (1) ⊗ · · · ⊗ u σ ( n ) ; u k ∈ C d . This defines a representation, π n : Sym( n ) → L( F d ) of the symmetric or permutation groupSym( n ) on n letters. The n th symmetric tensor product of C d ,( C d ) n = C d ⊗ C n Sym( n ) ⊂ C d ⊗ C n , is (defined to be) the subspace of all common fixed points of the unitaries U σ . The symmetricFock space, Sym( F d ) (we will shortly identify this with H d ) is then the direct sum of allsymmetric tensor products: Sym( F d ) := ∞ M n =0 ( C d ) n . Let N d be the unital additive semigroup or monoid of d -tuples of non-negative integers.By the universality property of the free unital semigroup F d , there is a unital semigroupepimorphism λ : ( F d , · ) → ( N d , +), the letter counting map which sends a given word α = i · · · i n ∈ F d to n = ( n , ..., n d ) ∈ N d where n k is the number of times the letter k appears in the word α . For any n ∈ N d , we define the symmetric monomial L n := X α ∈ F d ; λ ( α )= n L α , and it is then not difficult to verify that { e n := L n R n } is an orthogonal basis forSym( F d ) such that h L n , L m i F = δ n , m | n | ! n ! . M.T. JURY AND R.T.W. MARTIN
Here, and throughout, we use the standard notations | n | := n + ... + n d for n ∈ N d , and n ! := n ! · · · n d !. As shown in, e.g [10], S d := _ n ∈ N d L n = _ z ∈ B d ( I − Lz ∗ ) − , where W denotes norm-closed linear span and Lz ∗ := L z + ... + L d z d . It follows thatSym( F d ) = _ ( I − Lz ∗ ) − , and it is easily verified that the map( I − Lz ∗ ) − k z , is an onto isometry which sends e n = L n z n | n | ! n ! ∈ H d , where z n := z n · · · z n d d . For theremainder of the paper we will identify these two spaces and simply write H d ⊂ F d forsymmetric Fock space.2. Free Formal reproducing kernel Hilbert spaces
It will be useful to review the theory of free or non-commutative (NC) formal reproducingkernel Hilbert spaces (RKHS), as introduced in [21, 19]. This will allow us to define left andright free analogues of the commutative several-variable deBranges-Rovnyak spaces H ( b )associated to any b ∈ S d ( H ). If B is a left or right free lift of b , we will see that there isvery nice relationship and natural maps between the corresponding free and commutativedeBranges-Rovnyak spaces. Moreover the left or right deBranges-Rovnyak space of B will have a structure which is formally very similar to a commutative deBranges-RovnyakRKHS, and it will be fruitful to exploit this analogy with the commutative setting to obtaina non-commutative or ‘free’ extension of the Aleksandrov-Clark theory for the abelian Schurclass S d ( H ) developed in [10, 11].Any formal RKHS in the sense of [21] is essentially a classical RKHS on a finitely gener-ated unital semigroup (or monoid), M d (with d generators), where the reproducing kernel K ϕ,ϑ ; ϕ, ϑ ∈ M d is viewed as the formal power series coefficients of a ‘formal reproducingkernel’ in two formal variables. The key difference between classical RKHS theory overfinitely generated monoids and formal RKHS theory is the shift in focus from multipliers to formal multipliers : Given a discrete RKHS, H ( K ϕ,ϑ ), of functions on a finitely generatedmonoid M d , instead of the usual multiplier algebra, one can consider the convolution alge-bra of bounded convolution operators from H ( K ϕ,ϑ ) into itself. If one identifies elementsof the discrete RKHS H ( K ϕ,ϑ ) with formal power series indexed by M d , this convolutionalgebra can be viewed as the formal multiplier algebra , the algebra of formal power serieswhich multiply the formal RKHS into itself. We will primarily be interested in the case of F d , the free unital semigroup on d generators (the universal monoid on d generators). REE ALEKSANDROV-CLARK THEORY 9
Formal RKHS over F d . Let H be an auxiliary ‘coefficient’ Hilbert space. We willcall any positive kernel function c : F d × F d → L( H ) an operator-valued free coefficientkernel , and the associated formal power series K ( Z, W ) := X α,β ∈ F d Z α ( W ∗ ) β T c ( α, β ) ∈ L( H ) { Z, W ∗ } , is called a (positive) free kernel . Here Z = ( Z , ..., Z d ) and W ∗ = ( W ∗ , ..., W ∗ d ) are twosets of free (non-commuting) variables and given a word α = i i ...i n ; i k ∈ { , ..., d } , thetranspose of α is α T = i n ...i . In the above, we have also used the notation L( H ) { Z, W ∗ } for the linear space of all formal power series in the free variables Z, W ∗ with coefficientsin L( H ), and we will write K α,β := c ( α, β ) for the coefficient kernel corresponding to a freekernel K .A Hilbert space K is called a free RKHS of H -valued functions if any F in K can bewritten as a formal power series F ( Z ) = X α ∈ F d Z α F α ∈ H { Z } ; F α ∈ H in the free variable Z , and if for each α ∈ F d the linear coefficient evaluation map K ∗ α ∈ L( K , H ) , defined by K ∗ α F := F α , is bounded. We write K α ∈ L( H , K ) for the Hilbert space adjoint of this linear map. Thefree coefficient kernel for K is defined by the coefficients K α,β := K ∗ α K β ∈ L( H ) . The expression, K ( Z, W ) := X α,β ∈ F d Z α ( W ∗ ) β T K α,β , defines a positive free kernel, called the free reproducing kernel of K . That is, K α,β isnecessarily a positive kernel function in the classical sense on the discrete set F d , andclassical RKHS theory implies that there is a bijection between free kernel functions K andfree RKHS, H ( K ), of free formal power series with free reproducing kernels K . We write K := F ( K ) if K is a free RKHS with free kernel K . Note that for any β ∈ F d , K β ( Z ) = X α ∈ F d Z α K α,β ∈ L( H , F ( K )) { Z } . If F ( K ) is a free RKHS of H -valued functions (whose elements can be written as freeformal power series), we can define formal point evaluation maps : K ∗ W := X α ∈ F d W α K ∗ α ∈ L( F ( K ) , H ) { W } . Also define the formal adjoint of free power series termwise as:(2.1) K W := ( K ∗ W ) ∗ := X α ∈ F d ( W ∗ ) α T K α ∈ L( H , F ( K )) { W ∗ } . Then for any F ∈ F ( K ), K ∗ W F is defined termwise as K ∗ W F = X α ∈ F d W α K ∗ α F = X α ∈ F d W α F α = F ( W ) , and K ∗ Z K W = X α X β Z α ( W ∗ ) β T K ∗ α K β = X α X β Z α ( W ∗ ) β T K α,β = K ( Z, W ) . (2.2)These properties are formally analogous to properties of classical RKHS, and in many calcu-lations it will be easier to work with the formal point evaluation maps K W ∈ L( H , F ( K )) { W ∗ } in place of the bounded linear coefficient evaluation maps K α ∈ L( H , F ( K )). Remark 2.2.
Up to this point, no new theory has been introduced. Under the identificationof elements of a free RKHS of formal power series with their power series coefficients indexedby the free monoid F d , the concept of a free RKHS is equivalent to that of a classical RKHSover F d .As in classical RKHS theory, given any free RKHS F ( K ) of H − valued free power se-ries, there are naturally associated (formal) free (left and right) multiplier algebras. Thenoncommutativity of the unital free semigroup ( F d , · ) leads to two different notions of for-mal multipliers: left multipliers and right multipliers (equivalently left or right convolutionoperators). REE ALEKSANDROV-CLARK THEORY 11
A bounded linear map M : F ( k ) → F ( K ) between two free RKHS of H and J -valuedfunctions, respectively, is called a left free multiplier if there is a formal power series M ( Z ) := X α ∈ F d Z α M α ∈ L( H , J ) { Z } , so that M acts as left multiplication by M ( Z ): For any F ∈ F ( k ),( M F )( Z ) = M ( Z ) F ( Z ) = X α ∈ F d Z α M α X β Z β F β := X α,β Z αβ M α F β = X γ Z γ X αβ = γ M α F β . Similarly it is called a right multiplier if it acts as right multiplication by M ( Z ):( M F )( Z ) = M ( Z ) • R F ( Z ) . The above right product of formal power series is defined as M ( Z ) • R F ( Z ) = X α Z α M α ! • R X β Z β F β := X α,β Z βα M α F β = X γ Z γ X βα = γ M α F β . The above shows that left and right formal free multiplication can be defined in terms of(left or right) convolution of the coefficients:
Lemma 2.3.
If a bounded linear M : F ( k ) → F ( K ) acts as left or right multiplication by M ( Z ) = P α Z α M α ∈ L ( H , J ) { Z } then M ∗ K α = X β · γ = α k γ M ∗ β ∈ L ( H ; F ( k )) , or M ∗ K α = X γ · β = α k γ M ∗ β ∈ L ( H ; F ( k )) , respectively. The restatement of the above in terms of the formal point evaluation maps is again moreformally analogous to the classical theory:
Lemma 2.4. If M : F ( k ) → F ( K ) is a bounded left multiplier then M ∗ K Z = k Z M ( Z ) ∗ ∈ L ( H , F ( k )) { Z ∗ } . If it is a bounded right multiplier then M ∗ K Z = k Z • R M ( Z ) ∗ ∈ L ( H , F ( k )) { Z ∗ } . Analogues of classical RKHS results include:
Theorem 2.5.
A formal power series M ( Z ) = P α Z α M α ∈ L ( H , J ) { Z } defines a boundedleft free multiplier from F ( k ) into F ( K ) if and only if there is a B > so that X β ′ · γ ′ = α ′ X β · γ = α M β k γ,γ ′ M ∗ β ′ ≤ B K α,α ′ , as positive free coefficient kernels.In particular, F ( k ) is contractively contained in F ( K ) if and only if K α,β − k α,β is apositive free coefficient kernel. Equality holds in the above with B = 1 if and only if M isa co-isometric left multiplier. The same statements hold for right free multipliers if one reverses the order of theproducts of the free semigroup elements β, γ and β ′ , γ ′ . Again, this can be restated interms of formal point evaluation maps and free kernels: Theorem 2.6.
A formal power series M ( Z ) = P α Z α M α ∈ L ( H , J ) { Z } defines a boundedleft free multiplier from F ( k ) into F ( K ) if and only if there is a B > so that M ( Z ) k ( Z, W ) M ( W ) ∗ ≤ B K ( Z, W ) , as free formal positive kernels.Similarly it defines a bounded right free multiplier if and only if M ( Z ) • R k ( Z, W ) • R M ( W ) ∗ ≤ B K ( Z, W ) . In either case (right or left) multiplication by M ( Z ) is a co-isometry if and only if equalityholds with B = 1 and F ( k ) is contractively contained in F ( K ) if and only if K − k is apositive free kernel. Given two free RKHS, F ( k ) , F ( K ), we define the left and right free multiplier spaces ,Mult L ( F ( k ) , F ( K )), Mult R ( F ( k ) , F ( K )), as the spaces of all left and right free multipliersof F ( k ) into F ( K ). As in the classical, commutative theory, any left (right) free multiplier, F , defines a bounded linear multiplication map, M LF : F ( k ) → F ( K ) (or M RF in the rightcase), and under this identification, these multiplier spaces are W OT -closed. In the casewhere F ( K ) = F ( k ), we write Mult L ( F ( K )) := Mult L ( F ( K ) , F ( K )), for the unital free leftmultiplier algebra of F ( K ) (and similarly for the free right multiplier algebra). As observed REE ALEKSANDROV-CLARK THEORY 13 above, the free left and right multiplier algebras of a free RKHS F ( K ) can be equivalentlyviewed as (what could be called) the free left and right convolution algebras of the discreteclassical RKHS H ( K α,β ) corresponding to the free coefficient kernel K α,β on F d × F d .Our main motivation for considering the theory of free formal RKHS is to apply it tothe setting of the full Fock space, F d , over C d . The example below (from [21]) shows thatthe full Fock space can be naturally viewed as a free RKHS, the free Hardy space over d free variables. The WOT-closed unital operator algebras generated by the left and rightcreation operators, i.e. the left and right free Toeplitz algebras, are then naturally identifiedwith the left and right free multiplier algebras of this free RKHS. Example 2.7.
The full Fock space and the free Szeg¨o kernel.Any element f ∈ F d has the form f = X α ∈ F d f α L α f α ∈ C , where 1 denotes the vacuum vector and L is the left creation isometry. We can identify f with the formal power series f ( Z ) := X α Z α f α . Since f α = h L α , f i F , the coefficient evaluation vector ˆ k α is simply ˆ k α ( Z ) = Z α , and thefree coefficient kernel is: ˆ k α,β := D ˆ k α , ˆ k β E F = δ α,β . The corresponding free kernel is then:ˆ k ( Z, W ) = X α,β ∈ F d Z α ( W ∗ ) β T ˆ k α,β = X α,β ∈ F d Z α ( W ∗ ) β T δ α,β = X α ∈ F d Z α ( W ∗ ) α T . This is a free analogue of the Szeg¨o kernel for Drury-Arveson space: Indeed, replacing
Z, W ∗ with the commutative variables z, w ∗ ∈ B d yields:ˆ k ( z, w ) = X α ∈ F d z α ( w ∗ ) β = X n ∈ N d | n | ! n ! z n ( w ∗ ) n = 11 − zw ∗ = k ( z, w ) , the Szeg¨o kernel for Drury-Arveson space. It makes sense to view F d as the ‘free’ Drury-Arveson space or free several-variable Hardy space.2.8. Free deBranges-Rovnyak spaces.
Viewing F d or vector-valued F d ⊗ H as a freeRKHS, the left and right free Toeplitz algebras, L ∞ d and R ∞ d , i.e. the unital WOT-closedalgebras generated by the left and right free shifts or creation operators, are naturallyidentified with the left and right free multiplier algebras of F d [21, 19]: L ∞ d ≃ Mult L ( F d ); and R ∞ d ≃ Mult R ( F d ) . We will use the notation L d ( H , H ) := [Mult L ( F d ⊗ H , F d ⊗ H )] = [ L ∞ d ⊗ L( H , H )] , and R d ( H , H ) := [Mult R ( F d ⊗ H , F d ⊗ H )] , for the left and right (operator-valued) free Schur classes , the closed unit balls of the leftand right multipliers between vector-valued Fock spaces over C d . Since the left and rightfree Toeplitz algebras L ∞ d and R ∞ d are each others commutants, the space of left multipliersMult L ( F d ⊗ H , F d ⊗ H ) can also be identified as the spaces of bounded linear mapswhich intertwine the scalar right multiplier algebras R ∞ d ⊗ I H and R ∞ d ⊗ I H acting onvector-valued Fock spaces. In the case where H = H = H , we simply write L d ( H ) for L d ( H , H ) := [ L ∞ d ⊗ L( H )] .As in the commutative setting, any element B = B L ∈ L d ( H ) or B = B R ∈ R d ( H ) canbe used to define a positive free deBranges-Rovnyak kernel ˆ k B and corresponding left orright free deBranges-Rovnyak space H L ( B ) or H R ( B ): Example 2.9.
Free deBranges-Rovnyak spacesConsider vector-valued Fock space F d ⊗ H . As in the commutative setting, any formaloperator-valued power series B ( Z ) ∈ L( H , J ) { Z } is the the left or right free Schur class if REE ALEKSANDROV-CLARK THEORY 15 and only if ˆ k L ( Z, W ) := ˆ k ( Z, W ) − B ( Z )ˆ k ( Z, W ) B ( W ) ∗ ∈ L( H ) { Z, W ∗ } , or ˆ k R ( Z, W ) := ˆ k ( Z, W ) − B ( Z ) • R ˆ k ( Z, W ) • R B ( W ) ∗ ∈ L( H ) { Z, W ∗ } , are free positive kernel functions, respectively, where ˆ k is the free Szeg¨o kernel of F d ⊗ H [19, Theorem 3.1].The (left or right) free deBranges-Rovnyak space is then defined as H L ( B ) := F (ˆ k L )or H R ( B ) := F (ˆ k R ), depending on whether B is in the left or right free operator-valuedSchur class.As in the commutative case, H R ( B ) can be defined as a complementary range space [9]: H R ( B ) := M (cid:16)q I F d ⊗ J − M RB ( M RB ) ∗ (cid:17) . Namely, H R ( B ) = Ran (cid:18)q I − M RB ( M RB ) ∗ (cid:19) equipped with the inner product that makes q I − M RB ( M RB ) ∗ a co-isometry onto its range: if P is the orthogonal projection ontoKer (cid:18)q I − M RB ( M RB ) ∗ (cid:19) ⊥ , (cid:28)q I − M RB ( M RB ) ∗ h, q I − M RB ( M RB ) ∗ g (cid:29) B := h P h, g i F . In the above, M RB ∈ L( F d ⊗ H , F d ⊗ J ) is defined by right free multiplication by B ( Z )(assume B belongs to the right Schur class). A similar statement, of course, holds if B isin the left Schur class.To see that H := M (cid:18)q I − M RB ( M RB ) ∗ (cid:19) and H R ( B ) = F (ˆ k R ) are the same space, firstnote that by free RKHS theory, H R ( B ) is contractively contained in F d ⊗ J since ˆ k ⊗ I J − ˆ k R is a positive free kernel. As in [9, Section I-3], H is also contractively contained in F d ⊗ J ,and if ˆ k denotes the free (operator-valued) Szeg¨o kernel and f = q I − M RB ( M RB ) ∗ g ∈ H ,then h h, f ( Z ) i H = D ˆ k Z h, f E F = (cid:28)q I − M RB ( M RB ) ∗ ˆ k Z h, g (cid:29) F = D ( I − M RB ( M RB ) ∗ )ˆ k Z h, f E H . This shows that H is a free RKHS with point evaluation maps K Z := ( I − M RB ( M RB ) ∗ )ˆ k Z = ˆ k Z − M RB ˆ k Z • R B ( Z ) ∗ , and free kernel K ( Z, W ) := ˆ k ( Z, W ) − B ( Z ) • R ˆ k ( Z, W ) • R B ( W ) ∗ = ˆ k R ( Z, W ) . This proves that H = H R ( B ). Note that in the above ˆ k Z ∈ L( J , F d ⊗ J ) { Z ∗ } is a formalpower series with coefficients in L( J , F d ⊗ J ), and we define the action of M RB , ( M RB ) ∗ onsuch formal power series (as well as the above inner products of formal power series) bylinearity. Alternatively, instead of formal manipulations with free formal power series, onecan arrive at the same conclusions by repeating the above arguments with the coefficientmaps. 3. Relationship to Non-commutative function theory
Free non-commutative function theory provides an alternative and equivalent mathemat-ical framework for defining non-commutative deBranges-Rovnyak spaces associated to theleft and right free Schur classes. In particular, there is a bijection between free RKHS F ( K )with free kernels K , and functional non-commutative (NC) RKHS of free non-commutative(NC) functions defined on NC sets [22, Theorem 3.20]. In this section we briefly describe therelationship between these two theories as they pertain to our program. Our presentationwill follow [23, 22].One inspiration for free non-commutative function theory is Popescu’s free functionalcalculus for row contractions (and Popescu’s theory of free holomorphic functions) [24, 25,26]. Recall that A := A Ld denotes the left free disk algebra, the unital operator algebragenerated by the left free shift (the row isometery of left creation operators) on the fullFock space, F d over C d . Further recall that the free left multiplier algebra of F d = F (ˆ k ) is L ∞ d , the unital WOT-closed operator algebra generated by the left free shift, also called theleft free Toeplitz algebra. Similarly we define operator-valued extensions of these algebras:given an auxiliary coefficient Hilbert space H , we will abuse notation slightly and write A Ld ⊗ L( H ) and L ∞ d ⊗ L( H ) for the operator-valued left free disk algebra, and the left freeToeplitz algebra, respectively. To be precise, we write A Ld ⊗ L( H ) , L ∞ d ⊗ L( H ) in placeof the norm and WOT-closure of these algebraic tensor products. These algebras are thenorm, and WOT-closure, respectively, of the unital operator algebras generated by theoperator-valued left free shift L ⊗ I H acting on vector-valued Fock space F d ⊗ H .The operator algebras L ∞ d , R ∞ d are unitarily equivalent via the transposition unitary U T : F d → F d : Given an orthonormal basis { e k } of C d and corresponding left and rightcreation operators L k v = e k ⊗ v , R k v = v ⊗ e k on F d , a canonical orthonormal basis for F d is the set { e α := L α } α ∈ F d . The unitary U T is then defined by transposition of the index: U T L α U T e α = e α T , REE ALEKSANDROV-CLARK THEORY 17 and α T denotes the transpose of α ∈ F d defined previously: if α = i · · · i k , α T = i k · · · i .It is easy to check that U T L α = R α T U T , and it follows that L ∞ d ≃ R ∞ d are unitarily equivalent. For this reason, when it is notnecessary to distinguish between left and right, we will identify L ∞ d with R ∞ d , and simplyuse F ∞ d to denote the free Toeplitz algebra . Any F ∈ F ∞ d can then be identified with a(unitarily equivalent) transpose-conjugate pair F = ( F L , F R ) ∈ L ∞ d × R ∞ d . In terms offormal power series, if(3.1) F L ( Z ) = X α Z α F α , then F R ( Z ) = ( F L ( Z )) T = X α Z α T F α = X α Z α F α T . This defines a transpose map on free formal power series, F R = T ◦ F L .Any F ∈ L ∞ d ⊗ L( H ) has the ‘free Fourier series’ of equation (3.1) which is defined bycomputing [27]: F ( Z ) = X α ∈ F d Z α F α := X α ∈ F d ( L α F α := M LF (1 ⊗ I H ); F α ∈ L( H ) . Given any 0 ≤ r <
1, and any F ∈ L ∞ d ⊗ L( H ), one can check as in e.g. [20, Lemma3.5.2, Theorem 3.5.5], that the power series X α ∈ F d ( rL ) α ⊗ F α , converges in operator norm for F d ⊗ H . This shows that F r ( Z ) := X Z α r α F α ∈ A Ld ⊗ L( H ) , belongs to the (operator-valued) free left disk algebra and one can check as in [24, Propo-sition 4.2] that M LF r converges to M LF in the strong operator topology as r → − .It is important to note, however, that as in the case of Fourier series for the classical diskalgebra [6], the partial sums of the free Fourier series for F ∈ A Ld may not converge, evenin the strong or weak operator topologies [27]. Instead, any F ∈ L ∞ d (or more generally L ∞ d ⊗ L( H )) can be recovered from its free Fourier series by taking Ces`aro sums. Namely,given any F ∈ L ∞ d ⊗ L( H ), the N th Ces`aro sum of F , Σ N ( F ) ∈ L ∞ d ⊗ L( H ) is the averageof the first N partial sums of the free Fourier series of F . As shown in [27], for any N ∈ N ∪ { } , Σ N : L ∞ d ⊗ L( H ) → L ∞ d ⊗ L( H ) defines a completely contractive unitalmap (into free polynomials) so that Σ N ( F ) converges in the strong operator topology of L( F d ⊗ H ) to F . Results of Popescu [28, 24, 25] show that any F ∈ L ∞ d ⊗ L( H ) can be used to definea function on strict row contractions: If T ∈ (cid:0) L( J ⊗ C d , J ) (cid:1) then by the Popescu-vonNeumann inequality k p ( T , ..., T d ) k L ( J ) ≤ k p ( L , ..., L d ) k = k M Lp k L ( F d ) ; p ∈ L( H ) h L i , where L( H ) h L i = L( H ) h L , ..., L d i denotes the algebra of polynomials in the d free (non-commuting) variables L k with coefficients in L( H ). This inequality (and its matrix-valuedversion) shows that p ( L ) ∈ L( H ) h L i 7→ p ( T , ..., T d ) , defines a unital completely contractive algebra homomorphism which can be extended bycontinuity to L ∞ d ⊗ L( H ).This functional calculus is one of the inspirations for free non-commutative functiontheory [23, 25, 26]. Here is a brief introduction which is sufficiently general for our purposes:Let V = C d , a complex vector space, and consider the disjoint union V nc := ∞ a n =1 V n × n , V n × n := V ⊗ C n × n = C n × n ⊗ C d . Elements Z ∈ V n × n are viewed as bounded row operators on C n : Z = ( Z , ..., Z d ) : C n ⊗ C d → C n . Consider the non-commutative (NC) open unit ball Ω ⊆ V nc ,Ω := ∞ a n =1 Ω n ; Ω n := (cid:16) C n × n ⊗ C d (cid:17) , each Ω n is the set of all strict row contractions on C n . This set Ω is an example of what iscalled a non-commutative (NC) set [23] (it is closed under direct sums, and it is also bothleft and right admissable in the terminology of [23]).A function F : Ω ⊂ V nc → L( H ) nc = ` L( H ) n × n is called a non-commutative or freefunction if it has the two properties: F : Ω n → L( H ) n × n ; F is graded , and, if Z ∈ Ω n , W ∈ Ω m , and α ∈ C m × n obey αZ = W α, then αF ( Z ) = F ( W ) α ; F respects intertwinings . The free function F is called:(i) locally bounded if for any Z ∈ Ω n , there is a δ n > F is bounded on theball of radius δ n about Z ∈ Ω n . REE ALEKSANDROV-CLARK THEORY 19 (ii) analytic or holomorphic on Ω if F is locally bounded and Gˆateaux differentiable :For any Z ∈ Ω n and W ∈ V n × n , the Gˆateaux derivative of F at Z in the direction W :lim t → F ( Z + tW ) − F ( Z ) t = ddt F ( Z + tW ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 =: δF ( Z )( W ) , exists.By [23, Theorems 7.2 and 7.4], any locally bounded free function F is automatically analytic,and analyticity of F also implies that F has a certain power series representation (Taylor-Taylor series) with non-zero radius of convergence about any Z ∈ Ω (it also implies F is Fr´echet differentiable), see [23, Chapter 7]. Moreover, the results of [25, 26, 23] show,remarkably, that many classical results from complex analysis and several complex variableshave purely algebraic proofs that extend naturally to this setting.Let Hol(Ω) ⊗ L( H ) denote the algebra of all free holomorphic functions on the non-commutative (NC) ball Ω taking values in L( H ) nc . As in [25], we define the (operator-valued) free Hardy algebra , as the algebra of all uniformly bounded free holomorphic func-tions on this NC domain Ω taking values in L( H ) nc : H ∞ (Ω) := { F ∈ Hol(Ω) | k F k ∞ < ∞} , where the supremum norm of F over the NC unit ball is k F k ∞ := sup Z ∈ Ω k F ( Z ) k . By the results of [23, Chapter 7], any H ∈ H ∞ (Ω) ⊗ L( H ) has a power series representation: H ( Z ) = X α ∈ F d Z α H α := X Z α ⊗ H α ; Z ∈ Ω , H α ∈ L( H ) , which converges absolutely for any Z ∈ Ω, and uniformly on any closed NC ball Ω r := ` (Ω r ) n , (Ω r ) n := (cid:2) C n × n ⊗ C d (cid:3) r of radius 0 < r < Theorem 3.1. ( [25, Theorem 3.1] , [23] ) The map Φ : F ∞ d ⊗ L ( H ) → H ∞ (Ω) ⊗ L ( H ) defined by H ( L ) := X L α ⊗ H α ∈ L ( H ) { L } 7→ H ( Z ) := X Z α H α ∈ L ( H ) { Z } , is a unital completely isometric isomorphism. Recall that the above power series for H ( L ) is to be understood as the SOT-limit ofCes`aro sums. Remark 3.2.
Using the free functional calculus of Popescu, it is not difficult to verify thatΦ is injective, unital, and completely isometric. Surjectivity follows from approximating any H ∈ H ∞ (Ω) ⊗ L( H ) by the partial sums of its Taylor-Taylor series expansion about0 n ∈ Ω n [23, Chapter 7]. We will call H ∞ (Ω) the several-variable free Hardy algebra , andunder the above identification we will use the terms free Hardy algebra and free Toeplitzalgebra interchangeably. Remark 3.3.
In recent research, the theory of positive kernel functions and RKHS has alsobeen extended to the free function theory setting [22]. In particular, it can be shown thatthe class of all free formal RKHS is naturally isomorphic to the class of non-commutativereproducing kernel Hilbert spaces (NC-RKHS) [22, Theorem 3.20]. A NC-RKHS can beviewed as a sort of reproducing kernel Hilbert space of free or non-commutative functionson a NC set. In particular, one can naturally identify or view our free deBranges-Rovnyakspaces as NC-RKHS of this type. We have found, however, that the free extension of ourcommutative Aleksandrov-Clark theory from [10, 11], seems to carry over most naturallyusing the formalism of free RKHS. Namely, many of the theorems and proofs of this pa-per are formally identical (or very similar) to those of [11], upon replacing formal pointevaluation maps K Z with the point evaluation maps K z , z ∈ B d .4. Free Herglotz functions and Aleksandrov-Clark maps
In this section we define free Herglotz functions and construct the free Aleksandrov-Clarkmaps associated to any element of the free operator-valued Schur classes. Our calculationshere are a formal analogue of the approach in [11] for the commutative Schur class ofDrury-Arveson space. As in the previous section, consider the NC set Ω = ` Ω n , whereΩ n = (cid:0) C n × n ⊗ C d (cid:1) is the set of all strict row contractions on C n . In what follows weinitially focus on the left case, analogous results hold for the right case. Definition 4.1.
The free left Herglotz-Schur class , L + d ( H ), is the set of all free holomorphic L( H ) nc -valued functions H L ( Z ) ∈ L( H ) { Z } on the NC unit ball Ω such that the left freeHerglotz kernel :ˆ K L ( Z, W ) := 12 (cid:16) H L ( Z )ˆ k ( Z, W ) + ˆ k ( Z, W ) H L ( W ) ∗ (cid:17) ∈ L( H ) { Z, W ∗ } , is a positive formal free kernel.This expression for ˆ K L converges in operator norm for fixed Z, W ∈ Ω n , and this implies,in particular, that Re (cid:0) H L ( Z ) (cid:1) ≥ Z ∈ Ω n [22]. That is, H L ( Z ) is a bounded,accretive operator for any Z ∈ Ω. It then follows as in [8, Chapter IV.4], that H L ( Z ) + I is invertible, and that B LH ( Z ) := ( H L ( Z ) + I ) − ( H L ( Z ) − I ) ∈ L( H ) { Z } REE ALEKSANDROV-CLARK THEORY 21 is contraction-valued on the NC unit ball Ω so that B LH ∈ [ H ∞ (Ω) ⊗ L( H )] = L d ( H )belongs to the free left Schur class. Moreover, I − B LH ( Z ) = 2( H L ( Z ) + I ) is invertible forany Z ∈ Ω, and the free deBranges-Rovnyak kernel ˆ k L of B LH is given byˆ k L ( z, w ) = (cid:0) I − B LH ( Z ) (cid:1) ˆ K L ( Z, W ) (cid:0) I − B LH ( W ) ∗ (cid:1) . The free right Herglotz-Schur class, R + d ( H ), is defined similarly, and given H L ∈ L + d ( H ),it easy to see that the formal transpose maps L + d ( H ) onto R + d ( H ) and if we define H R := T ◦ H L , then B RH = T ◦ B LH .Conversely, let B = ( B L , B R ) ∈ L d ( H ) × R d ( H ) be a free Schur class transpose-conjugatepair. Motivated by the above, we will assume that any such B L , B R are non-unital in thesense that I − B L ( Z ) , I − B R ( Z ) are invertible for any fixed Z ∈ Ω. Given such a pair, B ,one can define a transpose-conjugate pair of free holomorphic functions H B = ( H LB , H RB )on Ω by H LB ( Z ) := ( I − B L ( Z )) − ( I + B L ( Z )); Z ∈ Ω , and similarly for H RB . The free Herglotz kernel for H LB is thenˆ K L ( Z, W ) = 12 (cid:16) H LB ( Z )ˆ k ( Z, W ) + ˆ k ( Z, W ) H LB ( W ) ∗ (cid:17) = ( I − B L ( Z )) − ˆ k L ( Z, W )( I − B L ( W ) ∗ ) − , where ˆ k L is the free left deBranges-Rovnyak kernel for H L ( B ). It follows that ˆ K L (andsimilarly ˆ K R ) are positive free kernels so that H B = ( H LB , H RB ) is a transpose-conjugatepair of free Herglotz-Schur functions on Ω. It is easy to verify that the maps B H B and H B H are compositional inverses and define bijections between the non-unital freeSchur classes and the free Herglotz-Schur classes. Remark 4.2.
The assumption that a free Schur pair B = ( B L , B R ) ∈ L d ( H ) × R d ( H ) benon-unital is not very restrictive. A simple argument combining the free Schwarz lemmafor free holomorphic functions on the NC unit ball Ω (see [25, Theorem 2.4]) with auto-morphisms of the unit ball of L( H ) shows that B ( Z ) is strictly contractive on the NC unitball Ω if and only if B (0) = B ∅ is a strict contraction (for 0 ∈ Ω n ), and this happens if andonly if b (0) = B L ∅ = B R ∅ is a strict contraction, where b ∈ S d ( H ) is the image of B L or B R under the symmetrization (quotient by the commutator ideal) map. We say B is strictlycontractive if this holds, and certainly any strictly contractive B is non-unital.It seems reasonable that the assumption that B be non-unital can be relaxed if one iswilling to allow H L , H R to take values in unbounded operators see [11, Remark 1.10]. Wewill avoid such complications and assume throughout that B is non-unital.Given any non-unital B = ( B L , B R ) ∈ L d ( H ) × R d ( H ), we define the left free Herglotzspace, H L, + ( H B ) := F ( ˆ K L ), as the free RKHS corresponding to the free left Herglotz kernel ˆ K L of H LB . The above relationship between the left free deBranges-Rovnyak and leftfree Herglotz kernels shows that there is a natural unitary multiplier from H L ( B ) onto H L, + ( H B ): Lemma 4.3.
Given any non-unital B ∈ L d ( H ) , formal left multiplication by I − B ( Z ) isan isometry, M L ( I − B ) , of the left free Herglotz space H L, + ( H B ) onto the left free deBranges-Rovnyak space H L ( B ) . The action of this isometry on formal point evaluation maps is: M L ( I − B ) ˆ K LW = ( M L ( I − B ) − ) ∗ ˆ K LW = ˆ k Lw ( I − B ( W ) ∗ ) − ∈ L ( H , H L ( B )) { W ∗ } . Given any fixed left free Herglotz function H L , define a map φ : A + A ∗ → L( H ) by φ ( I ) := Re ( H ∅ ) ≥ φ ( L α T ) ∗ := 12 H α ; α = ∅ , where the H α ∈ L( H ) are the coefficients of the formal power series for H L . Extend φ sothat it is self-adjoint and linear. It follows that H L ( Z ) = 2 X α Z α φ ( L α T ) ∗ − φ ( I ) , by definition. Let CP ( A ; H ) denote the set of all completely positive maps from A + A ∗ into L( H ) (we simply write A + A ∗ in place of its norm closure). Recall here that A := A Ld is the left free disk algebra. Proposition 4.4.
The free left Herglotz kernel of H L , ˆ K L ( Z, W ) , has the form ˆ K L ( Z, W ) = X α,β Z α ( W ∗ ) β T φ (( L α T ) ∗ L β T ) , and the map φ belongs to CP ( A ; H ) . It will be useful to first show that any positive element in A + A ∗ is the limit of ‘sums ofsquares’: Let C := [ A + A ∗ ] + , the positive norm-closed cone of the (norm-closed) operatorsystem A + A ∗ , and let C := [ A ∗ A ] + , i.e. C is the positive norm-closed cone of elementswhich are ‘sums of squares’: p ∈ C ⇒ p = X a ∗ k a k ; a k ∈ A . Lemma 4.5.
Any positive element of A + A ∗ is the norm-limit of sums of squares, i.e. , C = C .Proof. Suppose not. Then there is a positive p ≥ A + A ∗ so that p ∈ C \ C . By theMinkowski cone separation theorem, there is a real linear functional λ : C → R so that λ ( q ) ≥ q ∈ C but λ ( p ) < λ to a bounded complex linear functional on A + A ∗ in the usual way: If x is self-adjoint in A + A ∗ then x = p − q for p, q ∈ C . Then let Λ( x ) := λ ( p ) − λ ( q ), and REE ALEKSANDROV-CLARK THEORY 23 if x = r + is in A + A ∗ with r, s self-adjoint in A + A ∗ then define Λ( x ) := Λ( r ) + i Λ( s ).This is possible since A + A ∗ is a unital operator system so that any self-adjoint element in A + A ∗ can be written as the difference of elements of C (and the real and imaginary partsof any x ∈ A + A ∗ are also in the operator system). We will simply write λ in place of itsextension Λ to A + A ∗ .Define a quadratic form on A by: h a, b i λ := λ ( a ∗ b ) ∈ C ; a, b ∈ A . This is a positive quadratic form or pre-inner product on A , h a, a i λ = λ ( a ∗ a ) ≥ a ∈ A , since a ∗ a ∈ C . As in the usual Gelfand-Naimark-Segal (GNS) construction if N λ ⊂ A is the closed subspace of vectors of length zero with respect to h· , ·i λ , then this pre-innerproduct promotes to an inner product on A N λ , and we let H λ denote the Hilbert space completion of this inner product space.We can also define a GNS representation π λ : A → L( H λ ) in the usual way: π λ ( a )( b + N λ ) := ab + N λ . This is well-defined since N λ is a closed left A -module. It is not hard to see that π λ isa completely contractive and unital representation of A , and so it extends naturally to acompletely positive unital map on A + A ∗ . Since p ≥ π λ is positive, it follows that π λ ( p ) ∈ L( H λ ) is a positive operator. This produces the contradiction: h , π λ ( p )1 i λ = λ ( p ) < , and we conclude that C = C . (cid:3) Proof. (of Proposition 4.4 ) Let ˆ K := ˆ K L . We have that2 ˆ K ( Z, W ) = H L ( Z )ˆ k ( Z, W ) + ˆ k ( Z, W ) H L ( W ) ∗ = X α,β Z αβ ( W ∗ ) β T H α + X α,β Z α ( W ∗ ) α T β T H ∗ β = X γ,β Z γ ( W ∗ ) β T X αβ = γ H α + X α,γ Z α ( W ∗ ) γ T X βα = γ H ∗ β = X α,β ( Z ) α ( W ∗ ) β T X γβ = α H γ + X γα = β H ∗ γ , and this calculation shows that the coefficient kernel of the free positive kernel ˆ K is:ˆ K α,β := 12 X γβ = α H γ + X γα = β H ∗ γ . In particular it follows that ˆ K α, ∅ = 12 H α = φ ( L α T ) ∗ , by definition.Now suppose that α = λβ and observe that2 ˆ K λ · β,β = X γ · β = λ · β H γ + X γ · λ · β = β H ∗ γ = H λ = 2 ˆ K λ, ∅ . It follows that if α = λβ , the map φ obeys φ (( L α T ) ∗ L β T ) = φ ( L λ T ) ∗ , so that φ is well-defined on A + A ∗ . In order to arrive at the above equation, observe thatit was necessary that the transpose appears in the definition 2 φ ( L ∗ ) α = 2 φ ( L α T ) ∗ = H α .Since, for fixed α, β ∈ F d ,( L α T ) ∗ L β T = ( L γ T ) ∗ ; γ · β = αL γ T ; γ · α = β X γ ; γ · β = α ( L γ T ) ∗ + X γ ; γ · α = β L γ T , it follows that ˆ K α,β = φ (cid:16) ( L α T ) ∗ L β T (cid:17) . The fact that ˆ K α,β is a positive free coefficient kernel will imply that φ is completelypositive: Indeed, consider any element A ∈ A ⊗ C n × n of the form A = N X k =1 L α k ⊗ C k ; α k ∈ F d , C k ∈ C n × n . The set of all such finite sums is norm dense in A ⊗ C n × n . To show that φ is completelypositive, the (matrix-version of the) previous sums of squares lemma implies that it is REE ALEKSANDROV-CLARK THEORY 25 sufficient to show that φ ( n ) ( A ∗ A ) = ( φ ⊗ id n ) X k,j ( L α k ) ∗ L α j ⊗ C ∗ k C j ≥ , for all n ∈ N . The above can be written as φ ( n ) ( A ∗ A ) = N X k,j =1 φ (( L α k ) ∗ L α j ) ⊗ C ∗ k C j = N X k,j =1 ˆ K α Tk ,α Tj ⊗ C ∗ k C j = N X k =1 ˆ K α Tk ⊗ C k ! ∗ N X j =1 ˆ K α Tj ⊗ C j ≥ , and this proves that φ is completely positive. (cid:3) Consider the free Cauchy kernel ( I − ZL ∗ ) − := ∞ X k =0 ( ZL ∗ ) k = X α ∈ F d Z α ( L ∗ ) α ∈ L ∞ d { Z } = X α Z α ( L α T ) ∗ . (4.1)With this definition it follows thatˆ K L ( Z, W ) = φ (cid:0) ( I − ZL ∗ ) − ∗ ◦ ( I − W L ∗ ) − (cid:1) =: φ ( X Z α ( L α T ) ∗ )( X β W β ( L ∗ ) β ) ∗ = φ ( X Z α ( L α T ) ∗ ) X β ( W ∗ ) β T L β T = X Z α ( W ∗ ) β T φ (cid:16) ( L α T ) ∗ L β T (cid:17) . In the above, ∗ denotes the formal adjoint defined previously.With these definitions we also have that H L ( Z ) = φ (cid:0) ( I − ZL ∗ ) − ( I + ZL ∗ ) (cid:1) + i Im ( H ∅ ) , or equivalently, H L ( Z ) = φ (cid:0) I − ZL ∗ ) − − I (cid:1) + i Im ( H ∅ ) . This is the left free Herglotz formula, and it is clearly a non-commutative formal analogue ofthe classical Herglotz formula (1.1) from the introduction, as well as a direct free analogueof the commutative results for S d ( H ) obtained in [10, 11].This argument is reversible. Given any φ ∈ CP ( A ; H ) define a positive free kernelˆ K = ˆ K L and coefficient kernel ˆ K α,β byˆ K L ( Z, W ) := X Z α ( W ∗ ) β T ˆ K α,β = φ (cid:0) ( I − ZL ∗ ) − ∗ ◦ ( I − W L ∗ ) − (cid:1) , and(4.2) ˆ K α,β := φ (( L α T ) ∗ L β T ) . Complete positivity of φ ensures that this defines a positive coefficient kernel. If one defines H L ( Z ) := φ (cid:0) ( I − ZL ∗ ) − ( I + ZL ∗ ) (cid:1) , it follows that H L is a free holomorphic L( H )-valued function on the NC unit ball Ω, andone can calculate that(4.3) ˆ K L ( Z, W ) = 12 (cid:16) H L ( Z )ˆ k ( Z, W ) + ˆ k ( Z, W ) H L ( W ) ∗ (cid:17) . Indeed, ˆ K L ( Z, W ) = X α,β Z α ( W ∗ ) β T φ (( L α T ) ∗ L β T )= X α T ≥ β T X β + X β T ≥ α T X α φ (( L α T ) ∗ L β T ) − φ ( I )ˆ k ( Z, W ) . Consider the first sum. Since α T ≥ β T it follows that α T = β T γ T or α = γβ for some γ ∈ F d . This first sum can then be written as: X γ X β Z γβ ( W ∗ ) β T φ (cid:16) ( L β T γ T ) ∗ L β T (cid:17) = X γ,β Z γβ ( W ∗ ) β T φ ( L γ T ) ∗ = 12 (cid:0) H L ( Z ) + φ ( I ) (cid:1) ˆ k ( Z, W ) . The full calculation then establishes the formula (4.3). Since this is a positive free kernel,it follows that H L ∈ L + d ( H ) belongs to the left free Herglotz-Schur class.The entire above analysis can be repeated with right free Herglotz-Schur functions. Givena right free H R ∈ R + d ( H ) we can define φ ∈ CP ( A ; H ) by φ (( L α ) ∗ L β ) := ˆ K Rα,β . Then, ˆ K R ( Z, W ) = φ (cid:0) T ◦ ( I − ZL ∗ ) − ∗ ◦ T ◦ ( I − W L ∗ ) − (cid:1) , REE ALEKSANDROV-CLARK THEORY 27 where ∗ ◦ T ◦ ( I − W L ∗ ) − = ∗ ◦ T ◦ X W β ( L ∗ ) β = X ( W ∗ ) β T L β , and T is the formal transpose defined previously. Also note that ∗ ◦ T = T ◦ ∗ .In this right case we obtain the right Herglotz formula H R ( Z ) := T ◦ φ (cid:0) ( I − ZL ∗ ) − ( I + ZL ∗ ) (cid:1) + i Im ( H ∅ ) = T ◦ H L ( Z ) . Conversely, given φ ∈ CP ( A ; H ), one can define the right Herglotz function as above andit follows that any φ ∈ CP ( A ; H ) corresponds uniquely to a transpose-conjugate pair ofleft and right free Herglotz-Schur functions H = ( H L , H R ) ∈ L + d ( H ) × R + d ( H ). Thesearguments and formulas define bijections (modulo imaginary constant operators) betweentranspose-conjugate Herglotz-Schur pairs and completely positive maps on the free diskoperator system. In summary: Theorem 4.6.
There are bijections between the three classes of objects:(i) Transpose-conjugate pairs B = ( B L , B R ) ∈ L d ( H ) × R d ( H ) of non-unital free Schurclass functions.(ii) Transpose-conjugate pairs H = ( H L , H R ) ∈ L + d ( H ) × R + d ( H ) of free Herglotz-Schurfunctions.(iii) The positive cone CP ( A ; H ) of completely positive maps from the free disk operatorsystem A + A ∗ , A = A d , into L ( H ) .The bijection between free Schur class pairs and free Herglotz-Schur class pairs is given bythe maps B H B and H B H . The bijection (modulo imaginary constants) betweenthe free Herglotz-Schur classes and CP ( A ; H ) , H = ( H L , H R ) ∈ L + d ( H ) × R + d ( H ) ↔ φ ∈ CP ( A ; H ) , is given by the free Herglotz formulas: H LB ( Z ) := φ (cid:0) ( I − ZL ∗ ) − ( I + ZL ∗ ) (cid:1) + i Im ( H ∅ ) ; and H RB ( Z ) := T ◦ H LB ( Z ) . Again, observe that the above formula is formally analogous to the classical Herglotzrepresentation formula (1.1) for Herglotz functions on the disk. (It recovers the classicalformula in the scalar-valued, single-variable case if we identify AC measures on the unitcircle with positive linear functionals on the classical disk algebra.)
Definition 4.7.
We will use the notation µ B ∈ CP ( A ; H ) for the completely positive mapwhich corresponds uniquely to the transpose-conjugate pair B := ( B L , B R ) (equivalently to H B = ( H LB , H RB )) by the above theorem. The map µ B will be called the Aleksandrov-Clarkmap or non-commutative Aleksandrov-Clark measure of B . The free Cauchy transforms
As in [10, 11], given any φ = µ B ∈ CP ( A ; H ) one can construct a Gelfand-Naimark-Segal(GNS)-type space, F ( µ B ), and associated Stinespring representation π φ = π B : A + A ∗ → L( F ( µ B )). Here B = ( B L , B R ) is the unique transpose conjugate pair of free Schur classelements corresponding to φ . This construction relies on the semi-Dirichlet property of thefree disk algebra A [15]: A ∗ A ⊆ ( A + A ∗ ) −k·k . Briefly, given φ = µ B , consider the algebraic tensor product A ⊗ H , endowed with thepre-inner product h a ⊗ h , a ⊗ h i B := h h , µ B ( a ∗ a ) h i H . The fact that µ B ( a ∗ a ), and hence that this pre-inner product is well-defined relies on thesemi-Dirichlet property of A . If N B denotes the closed left A -module (or left ideal in A ) ofall vectors of length zero in this algebraic tensor product, then h· , ·i B promotes to an innerproduct on the quotient space A ⊗ H N B , and the Hilbert space completion of this inner product space will be denoted by F ( µ B ), the free Hardy space of µ B . The associated Stinespring representation is defined by a π B ( a )where π B ( a )( a ′ ⊗ h + N B ) := aa ′ ⊗ h + N B . The representation π B : A → L( F ( µ B )) is a unital completely isometric isomorphism whichis ∗ -extendible to a ∗ -representation of the Cuntz-Toeplitz C ∗ -algebra E := C ∗ ( A ) (and iswell-defined since N B is a left ideal). In particular it follows that π B ( L ) is a row-isometryon F ( µ B ) ⊗ C d . This yields the Stinespring dilation formula: µ B ( L α ) = [ I ⊗ ] ∗ B π B ( L ) α [ I ⊗ ] B ; α ∈ F d , where the bounded linear embedding [ I ⊗ ] B : H → F ( µ B ) is defined by[ I ⊗ ] B h := I ⊗ h + N B ∈ F ( µ B ) , and k [ I ⊗ ] B k = k µ B ( I ) k . This embedding is isometric if and only if µ B is unital.Recall that a CP map φ = µ B ∈ CP ( A ; H ) defines both a left and right free Herglotzspace with free kernels ˆ K L , ˆ K R , respectively. In what follows we consider the right case.The left case is, as usual, analogous. The formal point evaluation map ˆ K RZ is given by thefree formal series: ˆ K Z := X α ( Z ∗ ) α T ˆ K Rα . REE ALEKSANDROV-CLARK THEORY 29
Let B R be the right Schur class element defined by µ B . We define the free right Cauchytransform : ˆ C R : F ( µ B ) → H R, + ( H B ) , by(5.1) ˆ C R (cid:0) ∗ ◦ T ◦ [ I ⊗ ] ∗ B ( I − Zπ B ( L ) ∗ ) − (cid:1) := ˆ K RZ ∈ L( H ) { Z ∗ } . Expanding the above in free formal power series,ˆ C R X α ( Z ∗ ) α T π B ( L ) α [ I ⊗ ] B ! = ˆ K RZ , so that in terms of coefficient maps,ˆ C R ( π B ( L ) α [ I ⊗ ] B ) = ˆ K Rα . Remark 5.1.
Both the left and right hand sides of the above equation (5.1) are free powerseries in Z ∗ . To say that they are equal is to say that their coefficients are equal. We thenextend the action of ˆ C R to free power series by linearity.The free right Cauchy transform is an onto linear isometry since: (cid:0) ∗ ◦ T ◦ [ I ⊗ ] ∗ B ( I − Zπ B ( L ) ∗ ) − (cid:1) ∗ (cid:0) ∗ ◦ T ◦ [ I ⊗ B ] ∗ ( I − W π B ( L ) ∗ ) − (cid:1) = [ I ⊗ ] ∗ B T ◦ ( I − Zπ B ( L ) ∗ ) − ∗ ◦ T ◦ [ I ⊗ ] ∗ B ( I − W π B ( L ) ∗ ) − = [ I ⊗ ] ∗ B X α Z α π B ( L α ) ∗ X β ( W ∗ ) β T π B ( L ) β [ I ⊗ ] B = X α,β Z α ( W ∗ ) β T [ I ⊗ ] ∗ B π B ( L α ) ∗ π B ( L ) β [ I ⊗ ] B = X α,β Z α ( W ∗ ) β T µ B (cid:16) ( L α ) ∗ L β (cid:17) = ˆ K R ( Z, W ) , or, equivalently, (cid:16) ˆ C R π B ( L ) α [ I ⊗ ] B (cid:17) ∗ (cid:16) ˆ C R π B ( L ) β [ I ⊗ ] B (cid:17) = [ I ⊗ ] ∗ B π B ( L α ) ∗ π B ( L ) β [ I ⊗ ] B = ˆ K Rα,β = µ B (cid:16) ( L α ) ∗ L β (cid:17) (cid:16) = ˆ K Lα T ,β T (cid:17) . The weighted free right Cauchy transform ˆ F R : F ( µ B ) → H R ( B ) is then defined byˆ F R := ( I − B R ( Z )) • R ˆ C R , an onto isometry. As in Lemma 4.3 of Section 4, free right multiplication by ( I − B R ( Z ))is an isometry of the free right Herglotz space H R, + ( B ) onto the free right deBranges-Rovnyak space H R ( B ), and the inverse or Hilbert space adjoint of this isometry acts asfree right multiplication by ( I − B R ( Z )) − so that M R ( I − B ) = ( M R ( I − B ) − ) ∗ . It follows thatˆ F R (cid:0) ∗ ◦ T ◦ [ I ⊗ ] ∗ B ( I − Zπ B ( L ) ∗ ) − (cid:1) = ( M R ( I − B ) − ) ∗ ˆ K RZ = ˆ k Rz • R ( I − B ( Z ) ∗ ) − . (5.2)Similarly we can define the left free Cauchy and weighted Cauchy transforms, ˆ F L , ˆ C L byˆ C L (cid:0) ∗ ◦ [ I ⊗ ] B ( I − Zπ B ( L ) ∗ ) − (cid:1) := ˆ K LZ , or on coefficient maps as ˆ C L ( π B ( L ) α T ) ∗ [ I ⊗ ] B := ˆ K Lα , and ˆ F L := ( I − M LB ) ◦ ˆ C L . Proposition 5.2.
Let B := ( B L , B R ) ∈ L d ( H ) ⊗ R d ( H ) be a transpose conjugate pair.The onto isometry W T := ˆ F L ˆ F ∗ R : H R ( B ) → H L ( B ) acts by transposition: If F ( Z ) = P α Z α F α ∈ H R ( B ) then ( W T F )( Z ) = P α Z α T F α . The proof is easily verified, and omitted.6.
The Free Clark formulas
Assume that φ = µ B ∈ CP ( A ; H ) where B = ( B L , B R ) ∈ L d ( H ) × R d ( H ) is a transpose-conjugate pair of free (operator-valued) Schur multipliers. In this section we will developright free analogues of the Clark unitary perturbation formulas, the left case is analogous.Our approach and proof is a direct free analogue of the proof of the Clark intertwiningformulas for the commutative setting of Schur b ∈ S d ( H ). [11, Theorem 4.16, Section 4].A significant complication appears in the commutative Aleksandrov-Clark theory as soonas d >
1. Namely, in contrast to the classical single-variable theory [9], the deBranges-Rovnyak spaces H ( b ) for b ∈ S d ( H ) are generally not invariant for the adjoints of thecomponents of the Arveson d -shift on H d [17]. The appropriate replacement for the restric-tion of the backward shift in the several-variable theory is a contractive Gleason solution for H ( b ) [17, 18, 29, 30, 31]. Here, (see e.g. [11, Section 4]), a contractive Gleason solutionfor H ( b ) is a row contraction X : H ( b ) ⊗ C d → H ( b ) which obeys z ( X ∗ f )( z ) = f ( z ) − f (0); f ∈ H ( b ) , z ∈ B d , and which is contractive in the sense that XX ∗ ≤ I − k b ( k b ) ∗ . REE ALEKSANDROV-CLARK THEORY 31
Analogously, a map b : H → H ( b ) ⊗ C d is called a contractive Gleason solution for b ∈ S d ( H ) if z b ( z ) = b ( z ) − b (0); z ∈ B d , and if it is contractive in the sense that b ∗ b ≤ I H − b (0) ∗ b (0) . Observe that in the classical single-variable case, the unique contractive Gleason solutionsfor H ( b ) and b are given by X = S ∗ | H ( b ) and b = S ∗ b , where S is the shift on H ( D ).In contrast, as soon as d >
1, contractive Gleason solutions for H ( b ) and b are generallynon-unique (but they can be parametrized in a natural way, see [11, Section 4]).Every contractive Gleason solution for H ( b ) is determined by a contractive Gleasonsolution for b ∈ S d ( H ): Given any contractive Gleason solution X for H ( b ), there is acontractive Gleason solution b for b so that(6.1) X ∗ k bw = w ∗ k bw − b b ( w ) ∗ ; w ∈ B d , [11, Section 4] . Any contractive Gleason solution X for H ( b ) necessarily obeys: k bw = ( I − Xw ∗ ) − k b . Remarkably, the free theory is, in several ways, simpler and more closely parallels theclassical single variable theory. Any right free deBranges-Rovnyak space H R ( B ) for B =( B L , B R ) ∈ L d ( H ) × R d ( H ) is always invariant for L ∗ ⊗ I H , the adjoint of the left free shift(similarly H L ( B ) is invariant for R ∗ ⊗ I H ). Moreover, if one defines contractive (right) freeGleason solutions ˆ X R , B R for H R ( B ) and B as in the commutative setting, then these arealways unique and given by(6.2) ( ˆ X R ) ∗ := ( L ∗ ⊗ I H ) | H R ( B ) ; and B R := ( L ∗ ⊗ I H ) B R . (In the left case we obtain B L = ( R ∗ ⊗ I H ) B L and ( ˆ X L ) ∗ := ( R ∗ ⊗ I H ) | H L ( B ) .)Namely, a contractive Gleason solution for any right free deBranges-Rovnyak space H R ( B ) can be defined as a row-contraction ˆ X : H R ( B ) ⊗ C d → H R ( B ) such that Z (( ˆ X R ) ∗ F )( Z ) = F ( Z ) − F ∅ ; F ∈ H R ( B ) , and which is contractive in the sense thatˆ X R ( ˆ X R ) ∗ ≤ I − ˆ k R ∅ (ˆ k R ∅ ) ∗ . This definition is equivalent to ˆ k R ∅ = ( I − ˆ X R Z ∗ )ˆ k RZ , or,ˆ k RZ = ( I − ˆ X R Z ∗ ) − ˆ k R ∅ . Similarly, a contractive Gleason solution for B R is a map B R : H → H R ( B ) ⊗ C d whichobeys Z B R ( Z ) = B R ( Z ) − B R ∅ , and which is contractive in the sense that( B R ) ∗ B R ≤ I − ( B R ∅ ) ∗ B R ∅ . Remark 6.1.
Exactly as in the commutative setting, [11, Theorem 4.4], one can show thatif B is any contractive Gleason solution for B R thenˆ X ∗ ˆ k RW := ˆ k RW W ∗ − B • R B ( W ) ∗ , defines a contractive Gleason solution for H R ( B ). The transfer function theory of [19, seeRemark 4.4], shows that H R ( B ) , B R always have the unique contractive Gleason solutionsgiven by the formulas (6.2) above. Proposition 6.2.
The unique contractive Gleason solution B R : H → H R ( B ) ⊗ C d for B R is given by the formula B R = ( L ∗ ⊗ I H ) B R = ˆ F R π B ( L ) ∗ [ I ⊗ ] B ( I − B ∅ ) . Proof.
Write B := B R and let A ′ := ˆ F R π B ( L ) ∗ [ I ⊗ ] B . Then, Z A ′ ( Z ) = Z (ˆ k RZ ⊗ I d ) ∗ A ′ = (cid:16) ˆ F R π B ( L ) Z ∗ (ˆ F R ) ∗ ˆ k RZ (cid:17) ∗ ˆ F R [ I ⊗ ] B . Since ˆ F R := M R ( I − B R ) ˆ C R = ( M R ( I − B R ) − ) ∗ ˆ C R , it follows thatˆ F R [ I ⊗ ] B = ˆ k R ∅ ( I − B ∗∅ ) − . The bracketed term is thenˆ F R d X j =1 ( Z ∗ ) j π B ( L j )(ˆ C R ) ∗ ( M R ( I − B ) ) ∗ ˆ k RZ = ˆ F R d X j =1 ( Z ∗ ) j π B ( L j ) X α ( Z ∗ ) α T π B ( L α )[ I ⊗ ] B ! • R ( I − B ( Z )) ∗ = ˆ F R d X j =1 X α ( Z ∗ ) jα T π B ( L ) jα [ I ⊗ ] B • R ( I − B ( Z )) ∗ = ˆ F R X α ( Z ∗ ) α T π B ( L ) α [ I ⊗ ] B − [ I ⊗ ] B ! • R ( I − B ( Z )) ∗ = ˆ k RZ − ˆ k R ∅ ( I − B ∗∅ ) − ( I − B ( Z )) ∗ . REE ALEKSANDROV-CLARK THEORY 33
It follows that Z A ′ ( Z ) = (cid:16) ˆ k RZ − ˆ k R ∅ ( I − B ∗∅ ) − ( I − B ( Z )) ∗ (cid:17) ∗ ˆ k R ∅ ( I − B ∗∅ ) − = ˆ k R ( Z, ∅ )( I − B ∗∅ ) − − ( I − B ( Z ))( I − B ∅ ) − ( I − B ∅ B ∗∅ )( I − B ∗∅ ) − = ( I − B ( Z ) B ∗∅ )( I − B ∗∅ ) − − ( I − B ( Z ))( I − B ∅ ) − ( I − B ∅ B ∗∅ )( I − B ∗∅ ) − = ( I − B ( Z )) • R ˆ K R ( Z, ∅ ) −
12 ( I − B ( Z ))( H φ + H ∗ φ )= 12 ( I − B ( Z )) • R ( H B ( Z ) − H ∅ )= ( B ( Z ) − B ∅ )( I − B ∅ ) − . Hence A := A ′ ( I − B ∅ ) as defined above is a Gleason solution.To see that A is contractive note that if B R is a free lift of b ∈ S d ( H ),( A ∗ A ) ≤ ( I − B ∗∅ )[ I ⊗ ] ∗ B [ I ⊗ ] B ( I − B ∅ )= ( I − b (0) ∗ ) K b (0 , I − b (0))= I − b (0) ∗ b (0) = I − B ∗∅ B ∅ . By the uniqueness of the contractive Gleason solution for B R , A = B R = ( L ∗ ⊗ I H ) B R (Remark 6.1). (cid:3) Theorem 6.3. (right free Clark Intertwining) Let B = ( B L , B R ) ∈ L d ( H ) ⊗ R d ( H ) be atranspose conjugate pair of free Schur multipliers. The image of the adjoint of the row isom-etry π B ( L ) under the weighted right free Cauchy transform is a co-isometric perturbationof the restriction of L ∗ ⊗ I H to the (left free shift co-invariant) right deBranges-Rovnyakspace H R ( B ) : ˆ F R π B ( L ) ∗ (ˆ F R ) ∗ = L ∗ ⊗ I H | H R ( B ) + B R ( I − B ∅ ) − (ˆ k B ∅ ) ∗ , where B R = ( L ∗ ⊗ I H ) B R : H → H R ( B ) ⊗ C d is the unique contractive Gleason solutionfor B R . The left free Clark intertwining formulas are analogous and computed similarly. Theproof below is formally very similar to the Clark intertwining result for the commutativesetting of b ∈ S d ( H ), established in [11, Theorem 4.16, Section 4]. Remark 6.4.
As shown in [11], π B ( L ) is a Cuntz unitary (an onto row isometry) if andonly if the image b ∈ S d ( H ) of B under the Davidson-Pitts symmetrization (quotient) mapis quasi-extreme , i.e. if and only if H ( µ B ) := _ n ∈ N d ; n =0 L n ⊗ H = _ n ∈ N d L n ⊗ H =: H ( µ B ) ⊂ F ( µ B ) , (at least in the case where dim ( H ) < ∞ , see Remark 7.3). In the several-variable theory, H ( µ b ) and H ( µ b ) play the role of the classical analytic subspaces obtained as the closure ofthe analytic polynomials, and the closed linear span of the non-constant analytic monomialsin L ( µ b ) when d = 1 and µ b is an AC measure.If π B ( L ) is a Cuntz unitary, then the image of π B ( L ) ∗ under the weighted right Cauchytransform is a Cuntz unitary perturbation of the adjoint of the left free shift restricted tothe right free deBranges-Rovnyak space H R ( B ). This is a direct generalization of Clark’sclassical result [5] (Theorem 1.2), and we recover Clark’s result in the single-variable, scalar-valued case. Given any unitary U ∈ L( H ), it is not difficult to check that H R ( BU ∗ ) = H R ( B ). Applying the above result to BU ∗ for any such unitary U , yields the full U ( H )-parameter family of co-isometric Clark-type perturbations of the restriction of the adjointof the left free shift. Proof.
Let B := B R . Calculate on formal kernel maps:( L ∗ ⊗ I H )ˆ k RW = ( L ∗ ⊗ I H )ˆ k W − ( L ∗ ⊗ I H ) M RB ˆ k W • R B ( W ) ∗ = ˆ k W W ∗ − ( L ∗ ⊗ I H ) M RB ˆ k W • R B ( W ) ∗ . Observe that in terms of the formal power series, each L j ⊗ I H is a left multiplier so that L ∗ j ⊗ I H ˆ k W = ˆ k W W ∗ j , and then calculate,( L ∗ ⊗ I H ) M RB ˆ k W • R B ( W ) ∗ = ( L ∗ ⊗ I H ) M RB (ˆ k W − I ) • R B ( W ) ∗ + ( L ∗ ⊗ I H ) BB ( W ) ∗ = M RB ˆ k W W ∗ • R B ( W ) ∗ + B B ( W ) ∗ . In summary this shows(6.3) ( L ∗ ⊗ I H )ˆ k RW = ˆ k RW W ∗ − B B ( W ) ∗ , as expected, since L ∗ ⊗ I H | H R ( B ) is the unique contractive Gleason solution for H R ( B ).Compare this toˆ F R π B ( L ) ∗ (ˆ F R ) ∗ ˆ k BW = ˆ F R π B ( L ) ∗ X α = ∅ ( W ∗ ) α T π B ( L ) α [ I ⊗ ] B • R ( I − B ( W )) ∗ + ˆ F R π B ( L ) ∗ [ I ⊗ ] B ( I − B ( W ) ∗ )= ˆ F R π B ( L ) ∗ X α = ∅ ( W ∗ ) α T π B ( L ) α [ I ⊗ ] B • R ( I − B ( W )) ∗ + B ( I − B ∅ ) − ( I − B ( W )) ∗ , REE ALEKSANDROV-CLARK THEORY 35 where we have applied the previous proposition identifying B = B R with A to obtain thelast line above. It remains to calculateˆ F R π B ( L ) ∗ X α = ∅ ( W ∗ ) α T π B ( L ) α [ I ⊗ ] B = ˆ F R M j X β ( W ∗ ) β T W ∗ j π B ( L ) β [ I ⊗ ] B = ( M R ( I − B ) − ) ∗ ˆ C R (cid:0) ∗ ◦ T ◦ [ I ⊗ ] ∗ B ( I − W π B ( L ) ∗ ) − (cid:1) W ∗ = ( M R ( I − B ) − ) ∗ ˆ K RW W ∗ = ˆ k RW W ∗ • R ( I − B ( W ) ∗ ) − . In summary,(6.4) ˆ F R π B ( L ) ∗ (ˆ F R ) ∗ ˆ k BW = ˆ k RW W ∗ + B ( I − B ∅ ) − ( I − B ( W ) ∗ ) . Subtracting the expressions (6.3) and (6.4) yields: − ( L ∗ ⊗ I H )ˆ k RW + ˆ F R π B ( L ) ∗ (ˆ F R ) ∗ ˆ k RW = B B ( W ) ∗ + B ( I − B ∅ ) − ( I − B ( W )) ∗ . If we define T := B ( I − B ∅ ) − (ˆ k R ∅ ) ∗ : H R ( B ) → H R ( B ) ⊗ C d , then on point evaluation maps, T ˆ k RW = B ( I − B ∅ ) − (ˆ k R ∅ ) ∗ ˆ k RW = B ( I − B ∅ ) − ˆ k R ( ∅ ; W )= B ( I − B ∅ ) − ( I − B ∅ B ( W ) ∗ ) , and then ( T + ( L ∗ ⊗ I H ) − ˆ F R π φ ( L ) ∗ (ˆ F R ) ∗ )ˆ k RW = B (( I − B ∅ ) − ( I − B ∅ B ( W )) ∗ − B ( W ) ∗ − ( I − B ∅ ) − ( I − B ( W ) ∗ ) . The expression on the right evaluates to( I − B ∅ ) − ( I − B ∅ B ( W ) ∗ − ( I − B ∅ ) B ( W ) ∗ − I + B ( W ) ∗ )= ( I − B ∅ ) − ( I − B ∅ B ( W ) ∗ − B ( W ) ∗ + B ∅ B ( W ) ∗ − I + B ( W ) ∗ )= 0 , and this proves the Clark intertwining formulas. (cid:3) Relationship between the free and commutative theories
Recall the theory of non-commutative Aleksandrov-Clark measures for the commutativeseveral-variable operator-valued Schur class S d ( H ) [10, 11]. Let S = S d ⊂ A d = A be the (norm-closed) symmetrized operator subspace: S := _ n ∈ N d L n = _ z ∈ B d ( I − Lz ∗ ) − , where W denotes norm-closed linear span. Also recall that L n := X λ ( α )= n L α , where λ : ( F d , · ) → ( N d , +) is the unital letter-counting epimorphism. As in the free theoryof this paper, and as described in the introduction, there is a bijection between non-unital b ∈ S d ( H ), Herglotz-Schur class functions on B d , and completely positive (AC) maps µ b ∈ CP ( S ; H ), where CP ( S ; H ) is the positive cone of completely positive maps of S + S ∗ into L( H ). In particular the Herglotz representation formula in this setting is H b ( z ) = µ b (cid:0) ( I − Lz ∗ ) − ( I + Lz ∗ ) (cid:1) + i Im ( H b (0)) , which is formally very similar to our free Herglotz representation formulas of Theorem 4.6.The operator space S , like the full free disk algebra A , has the semi-Dirichlet property: S ∗ S ⊂ ( S + S ∗ ) −k·k , so that one can again apply a GNS-type construction to obtain the Hardy space of µ b , H ( µ b ), as the completion of the quotient of the algebraic tensor product S ⊗ H by vectorsof zero length with respect to the pre-inner product: h s ⊗ h , s ⊗ h i b := h h , µ b ( s ∗ s ) h i H . If φ = µ B ∈ CP ( A ; H ) is a completely positive extension of µ b , that the Hardy space H ( µ b ) of µ b embeds isometrically as a subspace H ( µ B ) ≃ H ( µ b ) of the free Hardy space F ( µ B ) of µ B . Corollary 7.1.
A free Schur class transpose-conjugate pair B = ( B L , B R ) ∈ L d ( H ) × R d ( H ) is a pair of free lifts of b ∈ S d ( H ) if and only if µ B ∈ CP ( A ; H ) is a completelypositive extension of µ b ∈ CP ( S ; H ) to the full free disk operator system A + A ∗ .Proof. If µ B extends µ b , then observe that H b ( z ) is obtained from H LB ( Z ) or H RB ( Z ) bysubstituting the commutative variable z ∈ B d in for Z . Hence b ( z ) is obtained from B L ( Z ) , B R ( Z ) in the same way. This substitution amounts to applying the Davidson-Pittssymmetrization map which is known to be a completely contractive unital epimorphism of L ∞ d ⊗ L( H ) or R ∞ d ⊗ L( H ) onto H ∞ d ⊗ L( H ) [13, Section 2].Conversely, if B L or B R is a free lift of b , then H LB ( Z ), or H RB ( Z ), evaluated at com-mutative z must equal H b ( z ). By the Herglotz representation formulas for the free and REE ALEKSANDROV-CLARK THEORY 37 commutative Herglotz-Schur classes, it follows that µ B (( I − Lz ∗ ) − ) = µ b (( I − Lz ∗ ) − ) , and this proves that µ B | S + S ∗ = µ b . (cid:3) Recall that any Schur class b ∈ S d ( H ), or µ b ∈ CP ( S ; H ) are said to be quasi-extreme if H ( µ b ) = H ( µ b ) where H ( µ b ) ⊂ H ( µ b ), is the several-variable analogue of the closedlinear span of the non-constant analytic monomials (see Remark 6.4). This quasi-extremeproperty is a natural analogue of the single-variable Szeg¨o approximation property as de-scribed in the introduction, and it is related to extreme points of the Schur class [16]. See[11] for several equivalent characterizations of this property. The free theory of this paperprovides yet another equivalent characterization. Corollary 7.2.
If a Schur class b ∈ S d ( H ) is quasi-extreme then it has a unique pair oftranspose-conjugate free lifts B = ( B L , B R ) ∈ L d ( H ) × R d ( H ) . The converse holds if H isfinite dimensional. Remark 7.3.
The converse holds provided that b is quasi-extreme if and only if µ b has aunique CP extension φ ∈ CP ( A ; H ). In [11, Proposition 4.17] this was proven for all finitedimensional H (and for a large class of b ∈ S d ( H ) with H separable [11, Proposition 4.14]).We expect b is quasi-extreme if and only if µ b has a unique extension, but the general resultfor separable H remains elusive at this time, see [11, Remark 2.1].7.4. The Free and commutative deBranges-Rovnyak spaces.
As before, B = ( B L , B R )is a transpose-conjugate pair of free Schur class functions B L ∈ L d ( H ), B R ∈ R d ( H ). Lemma 7.5.
The map C LH : H L ( B ) → H ( b ) defined by C LH ( I F − M LB ( M LB ) ∗ ) h = ( I H − M b M ∗ b ) h ; h ∈ H d ⊗ H , is a co-isometry onto H ( b ) with initial space [( I F − M LB ( M LB ) ∗ )( H d ⊗ H )] −k·k H L ( B ) . An analogous co-isometry C RH is defined for the right free deBranges-Rovnyak space. Proof.
Assume that B = B L and drop the superscript L , the same proof works for theright case. The proof follows from the definition of the deBranges-Rovnyak spaces ascomplementary range spaces: If h ∈ H d ⊗ H then k ( I F − M LB ( M LB ) ∗ ) h k H L ( B ) = k q I F − M LB ( M LB ) ∗ h k F = (cid:10) ( I F − M LB ( M LB ) ∗ ) h , h (cid:11) F = (cid:10) P H ( I F − M LB ( M LB ) ∗ ) P H h , h (cid:11) F = h ( I − M b M ∗ b ) h , h i H = k ( I − M b M ∗ b ) h k H ( b ) . In the above we used that H d ⊗ H is co-invariant for the left free multiplier M LB and that( M LB ) ∗ | H d ⊗ H = M ∗ b since B = B L is a left free lift of b . (cid:3) Recall that in the commutative theory, one defines Cauchy and weighted Cauchy trans-forms C b : H ( µ b ) → H + ( H b ) and F b : H ( µ b ) → H ( b ) by C b (( I − Lz ∗ ) − ⊗ h + N b ) = K bz h, and F b (( I − Lz ∗ ) − ⊗ h + N b ) = k bz ( I − b ( z ) ∗ ) − h ; F b = M ( I − b ) C b , and these define isometries onto the commutative Herglotz space H + ( H b ) and the deBranges-Rovnyak space H ( b ), respectively [11, Section 2.7]. Proposition 7.6.
Let B = ( B L , B R ) ∈ L d ( H ) × R d ( H ) be a transpose-conjugate pair offree lifts of b ∈ S d ( H ) . Then F b = C RH ˆ F R P = C LH ˆ F L P where P projects F ( µ B ) onto H ( µ B ) ≃ H ( µ b ) and ˆ F L , ˆ F R are the left and right weighted free Cauchy transforms ontothe left and right deBranges-Rovnyak spaces of B .Proof. We prove the right case, left is analogous. For z ∈ B d , we know that F b ( I − π B ( L ) z ∗ ) − [ I ⊗ ] B = k bz ( I − b ( z ) ∗ ) − . Compare the above toˆ F R (cid:0) ∗ ◦ T ◦ ( I − Zπ B ( L ) ∗ ) − [ I ⊗ ] B (cid:1) = ˆ k RZ • R ( I − B ( Z ) ∗ ) − . In particular, applying ˆ F R to ( I − π B ( L ) z ∗ ) − [ I ⊗ ] B amounts to substituting the commu-tative variables z in for Z in the above expression, whereˆ k Rz := ( I − M RB ( M RB ) ∗ )ˆ k z ( I − b ( z ) ∗ ) − , and ˆ k z := X α z α ˆ k α = X n ∈ N d z n ˆ k n . REE ALEKSANDROV-CLARK THEORY 39
In the above, recall that ˆ k α ( Z ) = Z α , and we define ˆ k n := P α ; λ ( α )= n ˆ k α . In particular,identifying H d with symmetric Fock space, we have that ˆ k z = k z ∈ L( H , H d ), so thatˆ F R ( I − π B ( L ) z ∗ ) − [ I ⊗ ] B = ( I − M RB ( M RB ) ∗ ) k z ( I − b ( z ) ∗ ) − ∈ Ker (cid:0) C RH (cid:1) ⊥ , and F b = C RH ˆ F R . (cid:3) Remark 7.7.
It is also easy to check that that the range of ˆ F R | H ( µ B ) is ( I − M RB ( M RB ) ∗ )( H d ⊗ H ), the initial space of the co-isometry C RH .7.8. Transfer function realizations.
As before, let B = ( B L , B R ) ∈ L d ( H ) × R d ( H )be a transpose-conjugate pair of free L( H )-valued Schur class functions. Recall that by[19], any B L ∈ L ∞ d ( H ) correpsonds uniquely to a (co-isometric, observable) canonicaldeBranges-Rovnyak colligation : U RdBR := " A RdBR B RdBR C RdBR D RdBR : " H R ( B ) H → " H R ( B ) ⊗ C d H , where, A RdBR := ( L ∗ ⊗ I H ) | H R ( B ) , B RdBR := L ∗ B R C RdBR := (ˆ k R ∅ ) ∗ , and , D RdBR := B R ∅ . The left Schur multiplier B L is then realized as the transfer function of U dBR by the Schurcomplement formula B L ( Z ) = D RdBR + C RdBR ( I − ZA RdBR ) − B RdBR , see [19, Theorem 4.3]. Note that A RdBR = ˆ X ∗ is (the adjoint of) the unique contractiveGleason solution for H R ( B ) and B RdBR = B R is our unique contractive Gleason solution for B R . This shows the (right) canonical deBranges-Rovnyak colligation for a left Schur classelement B L ∈ L d ( H ) is expressed in terms of operators on the right free deBranges-Rovnyakspace H R ( B ), see [19, Remark 4.5]. Similarly there is a canonical left colligation andtransfer function realization for B R using the left free deBranges Rovnyak space H L ( B ).In the commutative theory [17, 18] for Drury-Arveson space, any b ∈ S d ( H ) again al-ways has canonical (weakly co-isometric, observable) deBranges-Rovnyak transfer functionrealizations and colligations, but these are generally non-unique. Namely, a contraction, u dBR , is called a canonical deBranges-Rovynak colligation for b if it can be written in blockform as u dBR := " a dBR b dBR c dBR d dBR : " H ( b ) H → " H ( b ) ⊗ C d H , where d dBR := b (0), c dBR := ( k b ) ∗ , b dBR is a contractive Gleason solution for b , and X := a ∗ dBR is a contractive Gleason solution for H ( b ). As proven [18, Theorem 2.9, Theorem 2.10], given any contractive Gleason solution X for H ( b ), there is a contractiveGleason solution b for b so that the above colligation u dBR is a canonical deBranges-Rovnyak colligation (contractive, weakly co-isometric and observable). As in the free case, b ∈ S d ( H ) can be recovered from any such colligation u dBR with the transfer functionformula: b ( z ) = d dBR + c dBR ( I − z · a dBR ) − b dBR . In [11, Section 4], it was shown that there is a bijection between contractive Gleasonsolutions b : H → H ( b ) ⊗ C d for b ∈ S d ( H ) and row-contractive extensions D ⊇ V b ofa certain canonical row partial isometry V b on the commutative Herglotz space H + ( H b ).Namely, the map V b : H + ( H b ) ⊗ C d → H + ( H b ) defined by V b z ∗ K bz = K bz − K b ; z ∈ B d defines a partial isometry with initial space W z ∈ B d z ∗ K bz H . If V b ⊆ D : H + ( H b ) ⊗ C d → H + ( H b ) is any row-contractive extension of V b on H + ( H b ) (in the sense that D ( V b ) ∗ V b = V b ) then the formula(7.1) b [ D ] := U ∗ b D ∗ K b ( I − b (0)) , defines a contractive Gleason solution for b , and we let X [ D ] denote the contractive Gleasonsolution for H ( b ) corresponding to b [ D ] as in equation (6.1): X [ D ] ∗ k bz = z ∗ k bz − b [ D ] b ( z ) ∗ ; z ∈ B d . In the above, U b : H ( b ) → H + ( H b ) is the onto isometric multiplier of multiplication by( I − b ( z )) − . (We assume here that b ∈ S d ( H ) is non-unital, i.e. , I − b ( z ) is invertible for z ∈ B d and H b ( z ) takes values in bounded operators.) Finally, we set(7.2) u dBR [ D ] := " X [ D ] ∗ b [ D ]( k b ) ∗ b (0) ; D ⊇ V b , D : H + ( H b ) ⊗ C d → H + ( H b ) . Theorem 7.13 below will prove that any u dBR [ D ] is a canonical deBranges-Rovnyak colliga-tion for b , and that the map D u dBR [ D ] is surjective (neither of these facts is immediatelyobvious). Definition 7.9.
Given any non-unital b ∈ S d ( H ), let D ⊇ V b be a row contractiveextension of V b on H + ( H b ). Define the extension φ D ∈ CP ( A ; H ) of µ b ∈ CP ( S ; H ) by φ D ( L α ) := ( K b ) ∗ D α K b ∈ L( H ) . Such an extension will be called a symmetric extension .The fact that φ D ∈ CP ( A ; H ) extends µ b ∈ CP ( S ; H ) follows from: REE ALEKSANDROV-CLARK THEORY 41
Lemma 7.10. ( [11, Lemma 3.14] ) A row contraction D : K ⊗ C d → K on K ⊇ H + ( H b ) extends V b , V b ⊆ D , if and only if K bz = ( I − Dz ∗ ) − K b ; z ∈ B d . In the case where D = V b , φ D is called the tight extension of µ b . This was defined andstudied in [11, 10]. Since each φ D extends µ b , Corollary 7.1 implies that φ D = µ B [ D ] for aunique transpose-conjugate pair B [ D ] = ( B [ D ] L , B [ D ] R ) ∈ L d ( H ) × R d ( H ). Lemma 7.11.
Let D ⊆ V b and let φ D be the corresponding symmetric extension. Then π D ( L ) := π µ B ( D ) ( L ) is unitarily equivalent to the minimal isometric dilation of D and H ( φ D ) ≃ H ( µ b ) is co-invariant for π D ( L ) . This motivates the terminology symmetric extension (the symmetric subspace H ( φ D ) ⊆ F ( φ D ) is co-invariant for π D ( L )). The proof is as in [11, Proposition 3.7, Lemma 3.8]: Proof.
Let π D := π φ D be the GNS representation of A on F ( φ D ). Then T := π D ( L ) isa row isometry and H ( φ D ) = W n ∈ N d T n [ I ⊗ ] φ D H is cyclic for T . Let W be the minimalisometric dilation of D on K D ⊇ H + ( H b ). Since W, L are row isometries, for any α, β ∈ F d ,( L α ) ∗ L β = L λ β = αλ ( L λ ) ∗ α = βλ , and similarly for W . Hence, assuming say that β = αλ , φ D (( L α ) ∗ L β ) = φ D ( L λ )= ( K b ) ∗ W λ K b = ( K b ) ∗ ( W α ) ∗ W β K b . It follows that the map C D : F ( φ D ) → K D defined by C D T α [ I ⊗ ] φ D := W α K b , is an onto isometry (onto by minimality of W ) which extends the Cauchy transform C b of H ( φ D ) onto H + ( H b ). In particular, C D T α = W α C D , and since H + ( H b ) is co-invariantfor W , W ∗ | H + ( H b ) = D ∗ , it follows that H ( φ D ) is co-invariant for T = π D ( L ). (cid:3) This also yields the generalized Clark intertwining formulas:
Theorem 7.12.
Given any row contractive extension D of V b on H + ( H b ) , the weightedCauchy transform intertwines the co-isometry π D ( L ) ∗ with a perturbation of the adjoint ofthe contractive Gleason solution X ( D ) for H ( b ) : F b π D ( L ) ∗ | H ( µ b ) = (cid:16) X [ D ] ∗ + b [ D ]( I − b (0)) − ( k b ) ∗ (cid:17) F b . Proof.
The proof is exactly as in [11, Section 4.15], using that H ( µ b ) ≃ H ( φ D ) is co-invariant for π D ( L ). (cid:3) Given any contractive extension D ⊇ V b , and corresponding φ D ∈ CP ( A ; H ) extending µ b ∈ CP ( S ; H ) as above, we write U LdBR [ D ] , U RdBR [ D ] for the canonical deBranges-Rovnyakcolligations for the unique free Schur pair B [ D ] = ( B [ D ] L , B [ D ] R ) ∈ L d ( H ) × R d ( H )corresponding to the extension φ D by Corollary 7.1. Theorem 7.13.
Given any non-unital b ∈ S d ( H ) , let B = ( B L , B R ) ∈ L d ( H ) × R d ( H ) be a transpose-conjugate pair of free lifts of b . Let u dBR = " a dBR b dBR c dBR d dBR := " C H ⊗ I d I H U dBR " C H I H ∗ : " H ( b ) H → " H ( b ) ⊗ C d H , where U dBR = " A dBR B dBR C dBR D dBR is either the left canonical deBranges-Rovnyak colligationfor B R or the right colligation for B L . Then u dBR =: Ad C H ◦ U dBR is a canonicaldeBranges-Rovnyak colligation for b such that b dBR = C H B dBR is a contractive Glea-son solution for b , and a ∗ dBR = C H A ∗ dBR C ∗ H is the contractive Gleason solution for H ( b ) corresponding to b dBR : a dBR k bw = w ∗ k bw − b dBR b ( w ) ∗ ; w ∈ B d . This defines a surjective map, Ad C H , from canonical deBranges-Rovnyak colligations offree lifts of b onto canonical colligations for b . Every canonical colligation for b has theform u dBR [ D ] for a unique contractive D ⊇ V b (see equation 7.2) and the map Ad C H isa bijection when restricted to canonical colligation pairs of the form ( U LdBR [ D ] , U RdBR [ D ]) .A colligation pair ( U LdBR , U
RdBR ) corresponding to a free Schur class pair B = ( B L , B R ) isin the inverse image of u dBR [ D ] under Ad C H if and only if the compression of π B ( L ) to H ( µ B ) ≃ H ( µ b ) is equal to C ∗ b D C b . Remark 7.14.
By [11, Theorem 4.17], b ∈ S d ( H ) is quasi-extreme if and only if V b is a co-isometry, or equivalently if and only if b has a unique contractive (and necessarily extremal)Gleason solution b = b [ V b ]. Moreover, in this case X = X [ V b ] is the unique contractiveGleason solution for H ( b ) and this solution is extremal. It follows easily from this that b ∈ S d ( H ) is quasi-extreme if and only if u dBR = u dBR [ V b ] is the unique contractivecanonical deBranges-Rovnyak colligation for b and this colligation is an isometry. Proof.
Consider the right colligation case, let B = B R be any right free lift of b , we suppressthe superscript R . Let U dBR be the unique canonical co-isometric deBranges-Rovnyakcolligation for B . Given B dBR = B RdBR = ( L ∗ ⊗ I H ) B R consider b dBR := C RH B dBR . This REE ALEKSANDROV-CLARK THEORY 43 map b dBR : H → H ( b ) ⊗ C d is contractive in the sense of a Gleason solution: b ∗ dBR b dBR ≤ B ∗ dBR B dBR ≤ I − b (0) ∗ b (0) . Here, recall that B R ∅ = b (0). Moreover, b dBR = C H B dBR = C H ˆ F R π B ( L ) ∗ [ I ⊗ ] B ( I − b (0))= F b P H ( µ b ) π B ( L ) ∗ [ I ⊗ ] B ( I − b (0)) , where we have applied Proposition 7.6 in the above. Define a row contraction D on H + ( H b )by D ∗ K b = U b b dBR ( I − b (0)) − . If we can show that D ⊇ V b , then equation (7.1) and the results of [11, Section 4] willimply that b dBR = b [ D ] is a contractive Gleason solution for b . By definition, D ∗ K b = C b P H ( µ b ) π B ( L ) ∗ [ I ⊗ ] B , so that D = C b P H ( µ b ) π B ( L ) ∗ | H ( φ D ) . Indeed, anything else in H ( µ b ) is spanned by elements of the form( Lz ∗ )( I − Lz ∗ ) − ⊗ h, and the action of π B ( L ) ∗ on such elements is the same as that of ˆ V := C ∗ b ( V b ) ∗ C b . It followsthat D ⊇ V b so that b dBR = b [ D ] is a contractive Gleason solution for b .The corresponding Gleason solution ˆ X = A ∗ dBR obeysˆ X ∗ ˆ k RW = k RW W ∗ − B dBR B ( W ) ∗ , ˆ X ∗ = ( L ∗ ⊗ I H ) | H R ( B ) , let X := a ∗ dBR = C H ˆ XC ∗ H . Then, X ∗ k bw = C H ( L ∗ ⊗ I H )( I − M RB ( M RB ) ∗ ) k w = C H (cid:0) ( L ∗ ⊗ I H )( k w − k ) − ( L ∗ ⊗ I H ) M RB k w b ( w ) ∗ (cid:1) = C H (cid:0) w ∗ k w − ( L ∗ ⊗ I H ) M RB ( k w − k ) b ( w ) ∗ + ( L ∗ ⊗ I H ) M RB k b ( w ) ∗ (cid:1) = C H w ∗ ( I − M RB ( M RB ) ∗ ) k w + C H ( L ∗ ⊗ I H ) M RB k b ( w ) ∗ = w ∗ k bw + C H ( L ∗ ⊗ I H ) M RB k b ( w ) ∗ = w ∗ k bw + C H ( L ∗ ⊗ I H ) Bb ( w ) ∗ = w ∗ k bw + C H B dBR b ( w ) ∗ = w ∗ k bw + b dBR b ( w ) ∗ , and this shows that a ∗ dBR = X = X [ D ] is the contractive Gleason solution for H ( b )corresponding to b dBR = b [ D ]. Also note that c ∗ dBR = C H ˆ k R ∅ = k b . To prove that u dBR = u dBR [ D ] as defined in the theorem statement is a canonical deBranges-Rovnyakcolligation for b , it remains to show, by [18, Theorem 2.9], that u dBR is contractive. Since C H is a contraction, this is clear, and we conclude that Ad C H ( U dBR ) = u dBR [ D ].To prove that this map from canonical deBranges-Rovnyak colligations U dBR [ D ] for B todeBranges-Rovnyak colligations for b is onto, let u dBR be any canonical deBranges-Rovnyakcolligation for b . Since b dBR is a contractive Gleason solution for b , it follows that there isa contractive extension D ⊇ V b so that u dBR = " a dBR b [ D ]( k b ) ∗ b (0) . As described above, if φ D ∈ CP ( A ; H ) is the completely positive extension of µ b correspond-ing to D ⊇ V b , then φ D = µ B [ D ] for a unique pair of free lifts B [ D ] = ( B [ D ] L , B [ D ] R ).By Proposition 6.2, the unique contractive Gleason solution for H R ( B [ D ]) is B [ D ] R := ˆ F R π D ( L ) ∗ [ I ⊗ ] B [ D ] ( I − B ∅ ) , and as in the first part of the proof b := C RH B [ D ] R is a contractive Gleason solution for b .Since H ( µ b ) = H ( µ B [ D ] ) is co-invariant for π D ( L ), Proposition 7.6 implies that b = F b π D ( L ) ∗ [ I ⊗ ] b ( I − b (0)) . Again, by the first part of the proof b = b [ D ′ ] where the contractive extension D ′ ⊇ V b isdefined by ( D ′ ) ∗ K b = U b b ( I − b (0)) − = C b π D ( L ) ∗ [ I ⊗ ] b = D ∗ K b . (By Lemma 7.11.)This proves that D ′ = D , and as in the first part of the proof, it follows that the image of U L,RdBR [ D ] under conjugation by C H is u dBR [ D ], and that this is a canonical colligation for b . Since both u dBR [ D ] = " X [ D ] ∗ b [ D ]( k b ) ∗ b (0) , and u dBR = " a dBR b [ D ]( k b ) ∗ b (0) , are canonical colligations for b , the uniqueness result [32, Corollary 2.9], implies that a ∗ dBR = X [ D ], so that u dBR = u dBR [ D ], and Ad C H implements a bijection of canonical pairs( U LdBR [ D ] , U RdBR [ D ]) onto canonical colligations for b . (cid:3) REE ALEKSANDROV-CLARK THEORY 45
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