Blaschke-Singular-Outer factorization of free non-commutative functions
aa r X i v : . [ m a t h . F A ] F e b BLASCHKE-SINGULAR-OUTER FACTORIZATION OF FREENON-COMMUTATIVE FUNCTIONS
MICHAEL T. JURY, ROBERT T.W. MARTIN, AND ELI SHAMOVICH
Abstract.
By classical results of Herglotz and F. Riesz, any bounded analytic functionin the complex unit disk has a unique inner-outer factorization. Here, a bounded analyticfunction is called inner or outer if multiplication by this function defines an isometry or hasdense range, respectively, as a linear operator on the Hardy Space, H , of analytic functionsin the complex unit disk with square-summable Taylor series. This factorization can befurther refined; any inner function θ decomposes uniquely as the product of a Blaschkeinner function and a singular inner function, where the Blaschke inner contains all thevanishing information of θ , and the singular inner factor has no zeroes in the unit disk.We prove an exact analogue of this factorization in the context of the full Fock space,identified as the Non-commutative Hardy Space of analytic functions defined in a certainmulti-variable non-commutative open unit ball. Introduction
Fundamental structure results of Herglotz and Riesz (and later Beurling) [20, 45, 5] in the theoryof analytic functions in the complex unit disk, D , imply that any uniformly bounded analyticfunction, h , in D admits a Blaschke-Singular-Outer factorization : h = b |{z} Blaschke · s |{z} Singular · f |{z} Outer , where b is an inner Blaschke product , s is a singular inner and f is an outer function . Thereare several equivalent definitions of inner and outer functions in the unit disk. We will takeoperator-theoretic definitions as our starting point as these will most readily generalize to thenon-commutative (NC) multi-variable setting of the full Fock space over C d .The Hardy space, H ( D ), is the Hilbert space of analytic functions in the disk with square-summable Taylor series coefficients at the origin, and H ∞ ( D ) is the unital Banach algebra of alluniformly bounded analytic functions in D . The Hardy algebra, H ∞ = H ∞ ( D ) can be identifiedwith the multiplier algebra of H , the algebra of all functions in D which multiply H into itself.That is, if f ∈ H ∞ and g ∈ H , then f · g = h ∈ H , and multiplication by f defines a bounded multiplier , a bounded linear multiplication operator, M f , on H . One can then define f ∈ H ∞ to be inner if the multiplier M f is an isometry, or outer if M f has dense range. In particular,multiplication by the independent variable, z , defines an isometry on H , the shift , S = M z , sothat H ∞ = Alg( I, S ) − weak −∗ and this plays a central role in Hardy Space Theory [33, 49]. Blaschkeand singular inner functions can also be described in purely operator-theoretic terms. Namely, given First named author partially supported by NSF grant DMS-1900364. ny h ∈ H ∞ we define the shift-invariant space S ( h ) := (cid:26) f ∈ H (cid:12)(cid:12)(cid:12)(cid:12) fh ∈ Hol( D ) (cid:27) , of all H functions ‘divisible by h ’. Clearly g ∈ S ( h ) if and only if any zero of h is a zero of g with greater or equal multiplicity, and S ( h ) ⊇ Ran ( M h ). An inner function, θ ∈ H ∞ , is then a Blaschke inner or singular inner if S ( θ ) = θH , or S ( θ ) = H , respectively. Equivalently, θ is singular inner if it has no zeroes in the disk. These are not the usualstarting or historical definitions of Blaschke and singular inner functions, but they are equivalent,see [21, Chapter 5] or [49, Chapter III.1]. The goal of this paper is to extend the seminal Blaschke-Singular-Outer factorization of functions in H ∞ and H to elements of the NC Hardy spaces.Recent research has identified the full Fock space over C d ,(1.1) F d := ∞ M k =0 (cid:16) C d (cid:17) ⊗ k = C ⊕ C d ⊕ (cid:16) C d ⊗ C d (cid:17) ⊕ (cid:16) C d ⊗ C d ⊗ C d (cid:17) ⊕ · · · , with the Free or Non-commutative Hardy space , H ( B d N ), a canonical NC multi-variable analogue of H ( D ) [40, 42, 41, 11, 4, 22, 23]. Elements of H ( B d N ) are analytic matrix-valued functions definedin an NC multi-variable open unit ball, B d N , in several NC matrix-variables [52, 27, 1, 53, 54]:(1.2) B d N := ∞ G n =1 B dn ; B dn := (cid:16) C n × n ⊗ C × d (cid:17) . Here, we fix the row operator space structure in B dn . Namely, any d − tuple of n × n matrices, Z = ( Z , · · · , Z d ) ∈ B dn , can be viewed as a linear map from d copies of C n into one copy. The NCunit ball consists of the strict row contractions , i.e, the d − tuples satisfying ZZ ∗ = Z Z ∗ + · · · + Z d Z ∗ d < I. Elements of the full Fock space can be identified with power series in d non-commuting variableswith square-summable coefficients (see Section 2). That is, any f ∈ F d is a power series: f ( z ) := X α ∈ F d ˆ f α z α , where F d , the free monoid on d generators , is the set of all words in the d letters { , ..., d } , andgiven any word α = i · · · i n , i k ∈ { , ..., d } , z α := z i · · · z i n . At first sight this may appear to havelittle bearing to classical Hardy Space Theory and analytic function theory in the disk. However,foundational work of Popescu has shown that if Z := ( Z , · · · , Z d ) : H ⊗ C d → H is any strict rowcontraction on a Hilbert space, H , then the above formal power series for f converges absolutelyin operator norm when evaluated at Z (and uniformly on compacta) [40, 47]. It follows that any f ∈ F d can be viewed as a locally bounded free non-commutative function in the NC open unit ball, B d N [27]. That is, we can view F d as the NC Hardy space, H ( B d N ), the Hilbert space of all (analytic) ree NC functions in B d N with square-summable Taylor series coefficients. Non-commutative H ∞ , H ∞ ( B d N ) can then be defined as the unital Banach algebra of uniformly bounded free NC functionsin the NC open unit ball, and as in the single-variable setting, this can be identified (completelyisometrically [40, 47]) with the left multiplier algebra of H ( B d N ), the algebra of all free NC functionsin B d N which left multiply the NC Hardy space, H ( B d N ) into itself. Furthermore, again in exactanalogy with classical Hardy Space Theory, left or right multiplication by the independent NCvariables define isometries on the NC Hardy space: L k := M LZ k , R k := M RZ k , ≤ k ≤ d, and these have pairwise orthogonal ranges L ∗ k L j = I H δ k,j , so that the row operator: L :=( L , L , · · · , L d ) : H ( B d N ) ⊗ C d → H ( B d N ) is an isometry which we call the left free shift . TheNC Hardy algebra, H ∞ ( B d N ) is equal to Alg( I, L ) − weak −∗ , the left free analytic Toeplitz algebra .This algebra and its norm closed analogue were first studied by Popescu in [37] (see also [38]).Later they were also studied by Davidson and Pitts [11, 9, 10, 8], Arias and Popescu [3], and fur-ther by Popescu [35, 40, 42, 41]. In greater generality this setup was extensively studied by Muhlyand Solel [30, 31, 32].Popescu was the first to discover an NC analogue of the classical Beurling theorem for H ( B d N ) in[34, Theorem 2.2] (see also [36, Theorem 4.2] for the first instance of the inner-outer factorization).The theorem is also proven in [3, Theorem 2.1] and was later proven independently by Davidsonand Pitts [11, Corollary 2.2]. Inner-outer factorization of NC functions in H ( B d N ) or H ∞ ( B d N ) isan easy consequence of this; any H ∈ H ∞ ( B d N ) can be factored as H = Θ · F , where Θ is anNC inner (an isometric left multiplier) and F is an NC outer, i.e. M LF = F ( L ) has dense range.Equivalently F = M LF R − cyclic vector, and this second definition extends to F ∈ H ( B d N ). Inthis paper, we refine these results to include an exact NC analogue of the Blaschke-Singular-Outerfactorization. An NC Blaschke inner B ∈ H ∞ ( B d N ) will be an NC inner whose range is completelydetermined by its left ‘NC variety’ in the NC unit ball. An NC inner left multiplier S will besingular if S ( Z ) is invertible for any Z ∈ B d N . Theorem (NC Blaschke-Singular-Outer factorization, Theorem 5.10) . Every non-zero H ∈ H p ( B d N ) , p ∈ { , ∞} , can be factored as a product H = B · S · F for B, S ∈ H ∞ ( B d N ) , where B is an NCBlaschke inner with the same NC variety as H , S is an NC singular inner and F ∈ H p ( B d N ) is anNC outer function. The factors are unique up to scalars of unit modulus. The left NC variety of any NC Hardy space function is formally defined in Definition 3.2 below.Roughly speaking, the NC variety is the collection of directional zeroes in the sense of [18] and[19]. When d = 1, our NC Blaschke-Singular-Outer factorization theorem recovers the classicalfactorization with a new operator-theoretic proof, see Corollary 5.11.1.1. Outline.
Section 2 contains the necessary background on the NC unit ball, the NC Hardyspace, the NC Hardy algebra H ∞ ( B d N ), and its commutant — the algebra of right multipliers.In Section 3 we discuss the (left) NC varieties cut out as degeneracy loci of functions in the NC ardy spaces. Examples of computations of NC Blaschke inner and singular inner functions areprovided in Section 6. The main theorem stated above is proven in Section 5. Lastly, the appendixcontains a factorization result for NC idempotent-valued functions obtained while working on themain theorem and is of independent interest in our opinion.2. Preliminaries: Fock Space as the NC Hardy space
The free monoid, F d is the set of all words in d letters { , ..., d } . This is the universal monoid on d generators, with product given by concatenation of words, and unit ∅ , the empty word containingno letters. The Hilbert space of square summable sequences indexed by F d , ℓ ( F d ), and F d , thedirect sum of all tensor powers of C d , i.e. full Fock space over C d , are naturally isomorphic (seeequation 1.1). This isomorphism is implemented by the unitary map e i ··· i k e i ⊗ · · · ⊗ e i k , i k ∈ { , ..., d } , and e ∅ { e j } denotes the standard basis of C d , and 1 is the vacuum vectorof the Fock space (which spans the subspace C ⊂ F d ). Under this isomorphism the left free shiftsbecome the left creation operators on the Fock space which act by tensoring on the left with thestandard basis vectors of C d . In the sequel we identify the free square-summable sequences, ℓ ( F d )and the Fock space F d with the NC Hardy space, denoted by H ( B d N ): H ( B d N ) = f ∈ Hol( B d N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( Z ) = X α ∈ F d ˆ f α Z α , X | ˆ f α | < ∞ . Similarly, we will use the notation H ∞ ( B d N ) := Alg( I, L ) − weak −∗ , H ∞ ( B d N ) = ( f ∈ Hol( B d N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup Z ∈ B d N k f ( Z ) k < ∞ ) . Any element F ∈ H ∞ ( B d N ) is identified with the linear operator, F ( L ) := M LF , of left multiplicationby F ( Z ). As described in the introduction, H ∞ ( B d N ) can be identified with the left multiplieralgebra of H ( B d N ), and it immediately follows that H ∞ ( B d N ) ⊂ H ( B d N ). Any f ∈ H ( B d N ) is alocally bounded free non-commutative function in the sense of modern Non-commutative FunctionTheory [51, 27, 1]. That is, f respects the grading, direct sums and similarities which preserve itsNC domain, B d N . Any locally bounded free NC function (under mild, minimal assumptions on itsNC domain) is automatically holomorphic, i.e. it is both Gˆateaux and Fr´echet differentiable atany point Z ∈ B d N and has a convergent Taylor-type power series expansion about any point [27,Chapter 7].The right free shifts , R k = M RZ k are unitarily equivalent to the left free shifts L k = M LZ k via thetranspose unitary on ℓ ( F d ), U † , U † e α := e α † , where if α = i · · · i n ∈ F d , then α † := i n · · · i , its transpose.2.1. Fock space as an NC reproducing kernel Hilbert space.
The Hardy space, H ( D ) canbe equivalently defined using Reproducing Kernel Theory. Namely, H is the reproducing kernel ilbert space (RKHS) of the Szeg¨o kernel : k ( z, w ) := 11 − zw ∗ . As in the single-variable setting, the Free Hardy Space H ( B d N ) can be equivalently defined using(non-commutative) reproducing kernel theory [4]. All non-commutative reproducing kernel Hilbertspaces (NC-RKHS) in this paper will be Hilbert spaces of free NC functions in the NC unit ball, B d N . Any Hilbert space, H of NC functions in B d N , is a NC-RKHS if the linear point evaluation map, K ∗ Z : H → ( C n × n , tr n ) is bounded for any Z ∈ B dn . We will let K Z , the NC kernel map , denote theHilbert space adjoint of K ∗ Z , and, for any y, v ∈ C n , K { Z, y, v } := K Z ( yv ∗ ) ∈ H . Furthermore, given Z ∈ B dn , y, v ∈ C n and W ∈ B dm , x, u ∈ C m the linear map K ( Z, W )[ · ] : C n × m → C n × m , defined by ( y, K ( Z, W )[ vu ∗ ] x ) C n := h K { Z, y, v } , K { W, x, u }i H , is completely bounded for any fixed Z, W and completely positive if Z = W . This map is calledthe completely positive non-commutative (CPNC) kernel of H . As in the classical theory thereis a bijection between CPNC kernel functions on a given NC set and NC-RKHS on that set [4,Theorem 3.1], and if K is a given CPNC kernel on an NC set, we will use the notation H nc ( K )for the corresponding NC-RKHS of NC functions. The NC Hardy space, H ( B d N ), is then thenon-commutative reproducing kernel Hilbert space (NC-RKHS) corresponding to the CPNC Szeg¨okernel on the NC unit ball, B d N : K ( Z, W )[ · ] := X α ∈ F d Z α [ · ]( W α ) ∗ ; H ( B d N ) = H nc ( K ) . Adjoints of left multipliers have a familiar and natural action on NC kernel vectors:(2.1) F ( L ) ∗ K { Z, y, v } = K { Z, F ( Z ) ∗ y, v } . For our purposes, it will be convenient, as in [40], to view elements of the NC Hardy spaces asholomorphic (locally bounded) NC functions on all strict row contractions on a separable Hilbertspace. That is, we will add the infinite level to B d N :(2.2) B d ℵ := B d N G B d ∞ , where B d ∞ := (cid:16) C ∞×∞ ⊗ C × d (cid:17) , denotes the set of all strict row contractions on the separable Hilbert space C ∞ := ℓ ( N ), and C ∞×∞ := L ( ℓ ( N ). Here, and throughout, the notation C n × m denotes the n × m matrices withentries in C , so that C × d is a row with d entries. We will write C d in place of C d × . . NC Varieties
Let H ( Z ) be any free NC function in one of the NC Hardy spaces H ( B d N ) or H ∞ ( B d N ). The leftNC variety of H is the appropriate analogue of a variety in our NC multi-matrix-variable setting.The definition below is stated more generally for operator-valued left multipliers between vector-valued NC Hardy spaces. Let H , J be separable or finite-dimensional Hilbert spaces. We will write H ∞ ( B d N ) ⊗ L ( H , J ) in place of the weak operator topology (WOT) closure of this algebraic tensorproduct, viewed as left multiplication operators from H ( B d N ) ⊗ H into H ( B d N ) ⊗ J . Remark 3.1.
Any element F ( L ) ∈ H ∞ ( B d N ) ⊗ C n × m or H ∞ ( B d N ) ⊗ L ( J , H ) can be viewed as amatrix- or operator-valued function whose entries are bounded, free non-commutative functions in B d N or B d ℵ . Note, however, that F ( Z ), viewed as a function in B d N need not be NC in the sensethat it will generally not preserve direct sums. It can, however, be identified with a matrix-valuedNC function, e F ( Z ) ( i.e. e F does preserve direct sums, joint similarities and the grading) defined byconjugating F ( Z ) with appropriate basis permutation matrices [26, pp. 65–66], [43, p.38]. Definition 3.2.
Given any H ∈ H ∞ ( B d N ) ⊗ L ( H , J ) or H ∈ H ( B d N ) ⊗ H , the left singularity locus or left NC variety of H is:Sing( H ) := G n ∈ N ∪{∞} Sing n ( H )Sing n ( H ) := n ( Z, y ) ∈ B dn × C n (cid:12)(cid:12)(cid:12) y ∗ H ( Z ) ≡ o . The (left) singularity space of H is: S ( H ) := { h ∈ H ( B d N ) ⊗ J | y ∗ h ( Z ) ≡ ∀ ( Z, y ) ∈ Sing( H ) } . The singularity space of any such H (in vector-valued NC H or operator-valued NC H ∞ ) isclearly right shift invariant, and S ( H ) ⊇ Ran ( H ( L )) . In the above y ∗ H ( Z ) ≡ H ( L ) ∈ H ∞ ( B d N ) ⊗ L ( H , J ) and Z ∈ B dn , y ∈ C n means that h y ⊗ g, H ( Z ) x ⊗ h i C n ⊗ J = 0 , for any h ∈ H , g ∈ J , and any x ∈ C n . Remark 3.3.
Note that these varieties differ from the ones considered in [2, 47, 48] since thesevarieties correspond to a left ideal in the algebra of right multipliers and not to two-sided ideals.Similar varieties in the case of NC polynomials and NC rational functions were considered by Heltonand McCullough [18] and Helton, Klep and Putinar [19]. The projection onto the first coordinategives the variety of determinental zeroes considered, for example, in [16].
Remark 3.4.
Let H ∈ H p ( B d N ), p ∈ { , ∞} , and let π : F n ∈ N B dn × C n → B d N be the projection ontothe first coordinate. We claim that if π (Sing( H )) = B d N , then H ≡
0. In other words, if H is notidentically zero, then one cannot have det H ( Z ) = 0 for all Z ∈ B d N . Indeed, by [29, Theorem 5.7] he inner rank of H considered as a 1 × n n rank( H ( Z )) n (cid:12)(cid:12)(cid:12) Z ∈ a neighbourhood of 0 ∩ B dn o . This latter numberis less than 1 since det H ( Z ) = 0 for every Z ∈ B d N . Since the inner rank of H is either 1 or 0 weconclude that the inner rank of H is 0. However, this can only happen, if H ≡ Definition 3.5.
An NC left multiplier, H ( L ) ∈ H ∞ ( B d N ) ⊗ L ( H , J ), is:(1) inner , if H ( L ) is an isometry.(2) outer , if H ( L ) has dense range in H ( B d N ) ⊗ J .An element of Fock space, h ∈ H ( B d N ), is called NC outer if it is cyclic for the right shifts.The second definition of an NC outer h ∈ H ( B d N ) is equivalent to the first if H ∈ H ∞ ( B d N ).That is, if H ( L ) ∈ H ∞ ( B d N ), then h := H ( L )1 ∈ H ( B d N ) is NC outer if and only if H is NC outer.(In fact, any element h ∈ H ( B d N ) can be identified with a closed, densely-defined and generallyunbounded left multiplier, h ( L ) in the NC Smirnov class [24]. Under this identification, h ∈ H ( B d N )is NC outer if and only if h ( L ) has dense range.) Definition 3.6.
An NC inner (isometric) left multiplier Θ ∈ H ∞ ( B d N ) ⊗ L ( H , J ) is:(1) Blaschke if Ran (Θ( L )) = S (Θ).(2) singular if S (Θ) = H ( B d N ) ⊗ J . Remark 3.7.
A scalar NC inner S ∈ H ∞ ( B d N ) is singular if and only if it is pointwise invertible inthe NC unit ball, B d ℵ . Indeed, since the constant functions are in S ( S ), the singularity locus of S is empty. Thus for every 0 < r <
1, the operator S ( rL ) has dense range, i.e, it is an outer. ByTheorem 4.2 S ( rL ) is invertible and thus S ( Z ) is invertible for every Z ∈ B d ℵ .For simplicity, the following results are stated for scalar-valued NC left multipliers. These extendnaturally to operator-valued left multipliers between vector-valued NC Hardy spaces. Proposition 3.8.
Given any H ∈ H p ( B d N ) , p ∈ { , ∞} , Sing( H ) satisfies the following properties:(1) If ( Z, y ) , ( W, x ) ∈ Sing( H ) and c ∈ C , then ( Z ⊕ W, y ⊕ c · x ) ∈ Sing( H ) .(2) For S ∈ GL n and ( Z, y ) ∈ Sing( H ) , such that S − ZS ∈ B dn , we have that ( S − ZS, ( S ∗ ) − y ) ∈ Sing( H ) . Lemma 3.9.
Given any H ∈ H ∞ ( B d N ) or H ( B d N ) , the set S ( H ) is a closed, R − invariant subspaceand S ( H ) ⊥ = _ ( Z,y ) ∈ Sing( H ) K { Z, y, v } . Proof.
Clearly this is a subspace. If f ∈ S ( H ) then for any ( Z, y ) ∈ Sing( H ), we have that y ∗ ( R k f )( Z ) = y ∗ f ( Z ) Z k = 0 , so that R k f ∈ S ( H ). Observe that f ∈ S ( H ) if and only if0 = ( y, f ( Z ) v ) C n = h K { Z, y, v } , f i H ( B d N ) , or all ( Z, y ) ∈ Sing( H ) and all v ∈ C n . Hence if ( f n ) ⊂ S ( H ) and f n → f in norm, then for any( Z, y ) ∈ Sing( H ) so that Z ∈ B dn , and for any v ∈ C n ,( y, f ( Z ) v ) C n = h K { Z, y, v } , f i H ( B d N ) = lim n →∞ h K { Z, y, v } , f n i H ( B d N ) = lim ( y, f n ( Z ) v ) C n = 0 . This proves that S ( H ) is closed. (cid:3) Lemma 3.10. If Θ ∈ H ∞ ( B d N ) is NC inner then the kernels of the NC-RKHS (cid:0) Θ( L ) H ( B d N ) (cid:1) ⊥ have the form: K Θ { Z, y, v } := K { Z, y, v } − Θ( L ) K { Z, Θ( Z ) ∗ y, v } . Proof.
Easy to verify since I − Θ( L )Θ( L ) ∗ is the orthogonal projector onto (cid:0) Θ( L ) H ( B d N ) (cid:1) ⊥ . (cid:3) Corollary 3.11. If ( Z, y ) ∈ B dn × C n belongs to the singularity locus of an NC inner Θ( L ) , then (3.1) K Θ { Z, y, v } = K { Z, y, v } . Conversely, if v is cyclic for Alg(
I, Z ) and (3.1) holds, then ( Z, y ) is in the singularity locus.Proof. Clearly, since Θ( L ) is injective, we have that (3.1) holds if and only if K { Z, Θ( Z ) ∗ y, v } = 0.The latter holds if and only if for every f ∈ H ( B d N ) we have0 = h K { Z, Θ( Z ) ∗ y, v } , f i = h Θ( Z ) ∗ y, f ( Z ) v i . Hence, if (
Z, y ) is in the singularity locus, then the above equation holds. Conversely, if v is cyclic,then the set of all f ( Z ) v as f ranges over H ( B d N ) is a dense set and thus ( Z, y ) is in the singularitylocus. (cid:3)
Remark 3.12.
The above is not an if and only if statement in general. To see this consider Z = (cid:0) A B C (cid:1) ∈ B dn and set v = ( v ), and y = (cid:0) y (cid:1) , for some v , y = 0. Then for every f ∈ H ( B d N )we have f ( Z ) v = (cid:16) f ( A ) v (cid:17) and thus h K { Z, y, v } , f i H ( B d N ) = ( y, f ( Z ) v ) C n = 0 . Also for every f , f ( Z ) ∗ y = (cid:16) f ( B ) ∗ y (cid:17) and thus K { Z, f ( Z ) ∗ y, v } = 0 for every f . However, it neednot be the case that f ( B ) ∗ y = 0. This defect can be removed by relaxing our definition of NCvariety: Let the extended NC variety of H ∈ H ∞ ( B d N ) be the graded set:Sing ′ ( H ) := G n ∈ N ∪{∞} Sing ′ n ( H ) , where Sing ′ ( H ) := n ( Z, y, v ) (cid:12)(cid:12)(cid:12) Z ∈ B dn , y, v ∈ C n ; H ( Z ) ∗ y ⊥ Alg(
I, Z ) v o . The extended singularity space is then, S ′ ( H ) := { h ∈ H ( B d N ) | h ( Z ) ∗ y ⊥ Alg(
I, Z ) v ∀ ( Z, y, v ) ∈ Sing ′ ( H ) } . t is easily verified that this space is again R − invariant, closed, and that S ′ ( H ) ⊥ = _ ( Z,y,v ) ∈ Sing ′ ( H ) K { Z, y, v } . Moreover, with this definition, (
Z, y, v ) ∈ Sing ′ ( H ) if and only if K { Z, y, v } ∈ S ′ ( H ) ⊥ . Our originaldefinition is, however, fully justified by the NC Blaschke-Singular-Outer factorization theorem. Lemma 3.13.
An NC inner Θ is Blaschke if and only if Ran (Θ( L )) ⊥ = _ ( Z,y ) ∈ Sing n (Θ); v ∈ C n ; n ∈ N ∪{∞} K { Z, y, v } . Proof.
First any such Szeg¨o kernel vector is in Ran (Θ( L )) ⊥ by the last corollary. By definition, Θis Blaschke if the range of Θ( L ) is exactly the set of all f ∈ H ( B d N ) so that y ∗ f ( Z ) = 0 , ∀ ( Z, y ) ∈ Sing(Θ) . and this condition holds if and only if h K { Z, y, v } , f i = 0 , for all ( Z, y ) in this singularity locus. This, in turn, is equivalent to the corresponding set of NCSzeg¨o kernels spanning the orthogonal complement of the range of Θ( L ). (cid:3) NC Blaschke row-column factorization
By the NC inner-outer factorization theorem, any NC Hardy space function, H ∈ H p ( B d N ), p ∈ { , ∞} , in the NC unit ball factors uniquely as H ( L ) = Θ( L ) · F ( L ), where Θ ∈ H ∞ ( B d N ),Θ is NC inner and F ∈ H p ( B d N ) is NC outer [36, Theorem 4.2], [11, Corollary 2.2], [3, Theorem2.1]. (For the inner-outer factorization of operator-valued left multipliers between vector-valuedNC Hardy spaces, see [39, Theorem 1.7].) In this section, we therefore start with an NC innerfunction Θ ∈ H ∞ ( B d N ) and decompose it as the product of an NC Blaschke inner left row multiplierand an NC inner left column multiplier. Proposition 4.1.
Any NC inner Θ ∈ H ∞ ( B d N ) factors as Θ := B · S = ( B , ··· , B N ) S ...S N ! . where Ran ( B ( L )) = S (Θ) , Sing(Θ) = Sing( B ) , B is an NC Blaschke inner, all components B k ( L ) are inner with pairwise orthogonal ranges, and the column S is also inner.Proof. By [11, Theorem 2.1, Corollary 2.2] or [39, Theorem 1.7], there is a (row) inner B ( L ) : H ( B d N ) ⊗ C N → H ( B d N ) (where N ∈ N ∪ {∞} )), so that the R − invariant subspace S (Θ) = Ran ( B ( L )) . f f = Θ( L ) g ∈ Ran (Θ( L )), observe that for any ( Z, y ) ∈ Sing(Θ), that y ∗ f ( Z ) = y ∗ Θ( Z ) g ( Z ) = 0 , and it follows that Ran (Θ( L )) ⊆ Ran ( B ( L )). Since both B ( L ) , Θ( L ) are isometries, this impliesΘ( L )Θ( L ) ∗ ≤ B ( L ) B ( L ) ∗ so that by the Douglas Factorization Lemma [12], there is a contraction, S : H ( B d N ) → H ( B d N ) ⊗ C N so that Θ( L ) = B ( L ) · S, and Ran ( S ) ⊆ Ker( B ( L )) ⊥ . Moreover, R k Θ( L ) = B ( L )( R k ⊗ I N ) S = Θ( L ) R k = B ( L ) SR k , so that B ( L )(( R k ⊗ I n ) S − SR k ) = 0 , and since B ( L ) is an isometry ( R k ⊗ I n ) S − SR k = 0 . The weak −∗ closed unital algebra of the NC right shifts is the commutant of H ∞ ( B d N ) [11, Theorem1.2], and it follows that S = S ( L ) ∈ H ∞ ( B d N ) ⊗ C N is a column of left multipliers so thatΘ( L ) = B ( L ) S ( L ) = ( B ( L ) , ··· , B N ( L ) ) S ( L ) ...S N ( L ) ! . In the above, since Θ( L ) , B ( L ) are isometries, it follows that S ( L ) is also an isometry (or inner),and also each B k ( L ) is an isometry, so that the B k ( L ) must have pairwise orthogonal ranges. (cid:3) Our goal is to show that N = 1 so that both B and S are scalar NC inner functions, and it willfurther follow that S is a scalar NC singular inner. Theorem 4.2. If f ∈ H ( B d N ) is an NC outer, then f ( rL ) ∈ H ∞ ( B d N ) is invertible for ≤ r < . We will have several occasions to use the following concept of argument re-scaling map : Definition 4.3.
Given any r ∈ [0 , r : H ( B d N ) ⊗ H → H ( B d N ) ⊗ H be defined by:Φ r f = Φ r X α ∈ F d L α ⊗ ˆ f α := X α L α ⊗ r | α | ˆ f α =: f r . Similarly define ϕ r : H ∞ ( B d N ) ⊗ L ( H , J ) → H ∞ ( B d N ) ⊗ L ( H , J ) by ϕ r F ( L ) = F ( rL ).We sometimes write f r = f ( rL )1. If F ∈ H ∞ ( B d N ), then Φ r F ( L )1 = ϕ r ( F ( L ))1. Lemma 4.4.
For any < r ≤ , Φ r is a contractive, self-adjoint quasi-affinity. The map ϕ r is a completely contractive homomorphism for any r ∈ [0 , . If ϕ r : H ∞ ( B d N ) ⊗ L ( H ) → H ∞ ( B d N ) ⊗ L ( H ) , then it is also unital, and extends to a completely positive and unital map n the corresponding operator system. The map Φ r respects the module intertwining action of H ∞ ( B d N ) ⊗ L ( H , J ) : If F ( L ) ∈ H ∞ ( B d N ) ⊗ L ( H , J ) and f ∈ H ( B d N ) ⊗ H , then Φ r F ( L ) f = F ( rL ) f r . Lemma 4.5. If r ∈ [0 , , and f ∈ H ( B d N ) , then f ( rL ) := M L Φ r f ∈ H ∞ ( B d N ) .Proof. Write f = P ∞ n =0 f n , where each f n ∈ C { z , ..., z d } is a homogeneous NC polynomial of degree n . (This is the Taylor-Taylor series expansion of f at 0 ∈ B d .) Then f r = P r n f n , and the operatornorm of f r is k f r ( L ) k ≤ ∞ X n =0 r n k f n ( L ) k L ( H ( B d N )) | {z } = k f n ( L )1 k H B d N ) = ∞ X n =0 r n k f n k H , (the operator norm of any homogeneous free polynomial in L coincides with its Fock space norm), ≤ r − r · (cid:16)X k f n k H (cid:17) / = k f k H r − r . (cid:3) Proof. (of Theorem 4.2) Any NC outer F ∈ H ∞ ( B d N ) is necessarily pointwise invertible in the NCunit ball, B d N [24, Lemma 3.2], and this extends to any NC outer f ∈ H ( B d N ). (Otherwise there isa Z ∈ B dn and y ∈ C n so that f ( Z ) ∗ y = 0 and therefore K { Z, y, v } is orthogonal to the R − cyclicsubspace generated by f , for any v ∈ C n .) By the previous lemma, f ( rL ) ∈ H ∞ ( B d N ) is uniformlybounded. If f ( rL ) is not invertible, then it follows that f ( rZ ) − is not uniformly bounded in B d N , or,equivalently, f ( Z ) − is not uniformly bounded in r B d N . Since k f ( Z ) − k = k ( f ( Z ) ∗ ) − k , ( f ( Z ) ∗ ) − is not uniformly norm-bounded in r B d N , and it follows that we can find a sequence ( W ( n ) ) ⊂ r B d N , W ( n ) ∈ r B dm n , and y n ∈ C m n , k y n k = 1, so that k f ( W ( n ) ) ∗ y n k < n . We view each level C n as a subspace of C ∞ = ℓ ( N ) (the span of the first n standard basis vectors)so that each y n ∈ C ∞ . Let { e k } be the standard orthonormal basis for C ∞ , and choose a unitary U n so that U n y n = e . Then, since f ( Z ) is a free NC function, k f ( U n W ( n ) U ∗ n ) ∗ e k = k U n f ( W ( n ) ) ∗ U ∗ n e k = k f ( W ( n ) ) ∗ y n k → . It follows that we can assume, without loss in generality, that y n = e for every n ∈ N . That is,we can replace the uniformly bounded sequence of strict row contractions W ( n ) , with the sequence Z ( n ) := U n W ( n ) U ∗ n , and we set y = e = v . Since k Z ( n ) k ≤ r for every n ∈ N , it follows that the equence of NC Szeg¨o kernels (cid:0) K { Z ( n ) , e , e } (cid:1) is uniformly bounded in Fock space norm: k K { Z ( n ) , e , e }k H ( B d N ) = (cid:16) e , K ( Z ( n ) , Z ( n ) )[ E ] e (cid:17) C ∞ ≤ (cid:16) e , K ( Z ( n ) , Z ( n ) )[ I ] e (cid:17) ≤ k K ( Z ( n ) , Z ( n ) )[ I ] k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =0 Ad ( k ) Z ( n ) , ( Z ( n ) ) ∗ ( I ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ − r . In the above Ad
Z,Z ∗ denotes the completely positive map of adjunction by Z and Z ∗ ,Ad Z,Z ∗ ( P ) := Z P Z ∗ + · · · + Z d P Z ∗ d , and we used the fact that ZZ ∗ ≤ r I . Since this sequence of NC kernels is uniformly bounded, itfollows that there is a weakly convergent subsequence, (cid:0) K { Z ( k ) , e , e } (cid:1) (where say k = n k ) so that K { Z ( k ) , e , e } w → h ∈ H ( B d N ) . The vacuum coefficient of h is: h ∅ = h , h i H ( B d N ) = lim h , K { Z ( k ) , e , e }i H ( B d N ) = ( e , e ) C ∞ = 1 , and hence h = 0. However, for any NC polynomial p ∈ C { z , ..., z d } , consider: |h h, p ( R ) f i H ( B d N ) | = lim (cid:12)(cid:12)(cid:12) h K { Z ( k ) , e , e } , p ( R ) f i (cid:12)(cid:12)(cid:12) = lim (cid:12)(cid:12)(cid:12)(cid:16) e , f ( Z ( k ) ) p ( Z ( k ) ) e (cid:17) C ∞ (cid:12)(cid:12)(cid:12) ≤ lim k f ( Z ( k ) ) ∗ e kk p k H ∞ ( B d N ) (4.1) = 0 , by assumption. Since p ∈ C { z , ..., z d } was arbitrary and h = 0, we conclude f is not R − cyclic,contradicting the assumption that f ∈ H ( B d N ) is NC outer. (cid:3) Corollary 4.6.
Given any H ∈ H p ( B d N ) , p ∈ { , ∞} , if H = Θ · F is the inner-outer factorizationof H , then Sing( H ) = Sing(Θ) .Proof. We have that (
Z, y ) ∈ Sing( H ) if and only if y ∗ H ( Z ) = y ∗ Θ( Z ) F ( Z ) = 0 . ince F is outer, it is pointwise invertible in B d ℵ by the previous theorem, so that the above happensif and only if ( Z, y ) ∈ Sing(Θ). (cid:3)
Corollary 4.7.
For every < r < there is an H r ( L ) ∈ H ∞ ( B d N ) ⊗ C × N so that B ( rL ) = Θ( rL ) H r ( L ) , and H r ( L ) S ( rL ) = I H ( B d N ) . In particular, the NC column-inner, S ( L ) , is pointwise left invertible in the NC unit ball, B d ℵ .Proof. We have that Θ( L ) = B ( L ) S ( L ). For any 0 < r <
1, let,Θ( rL ) = Γ r ( L ) F r ( L ) , be the inner-outer factorization of Θ( rL ). Fix 0 < r < < s < s > r . Then,if 0 < t < stL ) = Γ s ( tL ) F s ( tL ) , where now F s ( tL ) is an invertible left multiplier by Theorem 4.2 so thatΓ s ( tL ) = Θ( stL ) F s ( tL ) − . By definition of B ( L ), it follows that if ( Z, y ) ∈ Sing(Θ) so that y ∗ Θ( Z ) = 0 , then necessarily, y ∗ B ( Z ) = 0 , and this shows that Ran (Θ( Z )) ⊥ ⊆ Ran ( B ( Z )) ⊥ , for any Z ∈ B dn , n ∈ N ∪ {∞} . In particular, for any 0 < r <
1, taking Z = rL ,Ran ( B ( rL )) −k·k ⊆ Ran (Θ( rL )) −k·k = Ran (Γ r ( L )) . Applying Douglas Factorization and using that H ∞ ( B d N ) is the commutant of the algebra of rightmultipliers [11, Theorem 1.2], it again follows that there is a bounded left row multiplier G r ( L ) sothat B ( rL ) = Γ r ( L ) G r ( L ) , and finally, B ( stL ) = Γ s ( tL ) G s ( tL )= Θ( stL ) F s ( tL ) − G s ( tL ) | {z } =: ˇ H s ( tL ) . In particular, since we fixed s > r , we can choose 0 < t < st = r , and B ( rL ) = Θ( rL ) H r ( L ) , where H r ( L ) := ˇ H r/t ( tL ) ∈ H ∞ ( B d N ) ⊗ C × N . This proves the existence of H r . Since B ( rL ) = Θ( rL ) H r ( L ), and Θ( rL ) is injective [11, Theorem1.7], it follows that Ker( B ( rL )) = Ker( H r ( L )). or any 0 < r <
1, Θ( rL ) = B ( rL ) S ( rL )= Θ( rL ) H r ( L ) S ( rL ) . Again, since Θ( rL ) is injective, it follows that H r ( L ) S ( rL ) = I H ( B d N ) . (cid:3) Corollary 4.8.
The matrix-valued left multiplier E r ( L ) := S ( rL ) H r ( L ) ∈ H ∞ ( B d N ) ⊗ C N × N , is idempotent. For any < r < , Ran ( I − E r ( L )) = Ker( E r ( L )) = Ker( B ( rL )) = Ker( H r ( L )) . For any < r, s < , H r · s ( L ) = H r ( sL ) and E r · s ( L ) = E r ( sL ) .Proof. If we define E r ( L ) := S ( rL ) H r ( L ), then E r ( L ) E r ( L ) = S ( rL ) H r ( L ) S ( rL ) | {z } = I H B d N ) H r ( L ) = E r ( L ) , proving that every E r is idempotent. Also, B ( rL ) = Θ( rL ) H r ( L )= B ( rL ) E r ( L ) , and it follows that the idempotent e r ( L ) := ( I H ( B d N ) ⊗ I N ) − E r ( L ) , takes values in the kernel of B ( rL ). Conversely, consider E r ( L ) = S ( rL ) H r ( L ). Clearly, Ker( H r ( L )) ⊆ Ker( E r ( L )) = Ran ( I − E r ( L )), and on the other hand if E r ( L ) x = 0 then0 = H r ( L ) E r ( L ) x = H r ( L ) S ( rL ) | {z } = I H r ( L ) x, so that Ker( B ( rL )) = Ker( H r ( L )) = Ker( E r ( L )) = Ran ( I − E r ( L )).Since B ( rL ) = Θ( rL ) H r ( L ), it follows thatΘ( rsL ) H r ( sL ) = ϕ s (Θ( rL ) H r ( L ))= ϕ s ( B r ( L )) = B ( rsL )= Θ( rsL ) H rs ( L ) . It follows that Θ( rsL ) ( H rs ( L ) − H r ( sL )) = 0 , and since Θ( rsL ) is injective, H r ( sL ) = H rs ( L ). Then, by definition of E r ( L ), E rs ( L ) = S ( rsL ) H rs ( L ) = S ( rsL ) H r ( sL ) = E r ( sL ) . Remark 4.9.
By [11, Corollary 1.8], the algebra H ∞ ( B d N ) contains no non-trivial idempotents.This result can be extended in a natural way to H ∞ ( B d N ) ⊗ C N × N to show that any NC idempotent E ∈ H ∞ ( B d N ) ⊗ C N × N factors as: E ( L ) = T ( L ) − ( I ⊗ P ) T ( L ) , where T ( L ) ∈ H ∞ ( B d N ) ⊗ C N × N is invertible and P ∈ C N × N is a fixed projection, see Appendix A. Remark 4.10.
Define operator-valued functions in B d ℵ by H ( Z ) := H r ( Z/r ) , and E ( Z ) := E r ( Z/r ) , where if k Z k = s < r is any value so that 0 < s < r <
1. This is well-defined since if0 < s = k Z k < r < t <
1, then H ( Z ) = H r ( Z/r ) = H t · r/t ( Z/r ) = H t ( Z/t ) . Then
H, E can be identified with operator-valued free NC functions in B d ℵ (see Remark 3.1), andthey are uniformly bounded on balls r B d ℵ of radius r < NC Blaschke-Singular-Outer Factorization
Consider the net of operator-valued left multipliers B ( rL ) ∈ H ∞ ( B d N ) ⊗ C × N for 0 < r ≤ R − invariant subspaces M r := Ran ( B ( rL )) −k·k , and let Q r denote the orthogonalprojections onto these spaces. Recall then, that P ⊥ r := R ( Q r ⊗ I d ) R ∗ , is the projection onto the range of the row isometry R | M r ⊗ C d , and that the wandering space of M r is defined to be the subspace: W r := M r ⊖ R M r ⊗ C d , with orthogonal projector P r := Q r − R ( Q r ⊗ I d ) R ∗ . Elements of W r = Ran ( P r ) are called wandering vectors , and if { Ω r ; k } N r k =1 is an orthonormal basisof wandering vectors then,Ω r ( L ) := (Ω r ;1 ( L ) , · · · , Ω r ; N r ( L )) ; Ω r ; k ( L ) := M L Ω r ; k , is a left-inner row multiplier with Ran (Ω r ( L )) = M r . We will call N r = dim( W r ) , the wandering dimension of M r . We then have the NC inner-outer factorization: B ( rL ) = Ω r ( L ) F r ( L ); Ω r ( L ) ∈ H ∞ ( B d N ) ⊗ C × N r , F r ( L ) ∈ H ∞ ( B d N ) ⊗ C N r × N , here F r ( L ) := Ω r ( L ) ∗ B ( rL ) is NC left outer for every 0 < r ≤
1. (Here, note that the DouglasFactorization Lemma implies the existence of a bounded linear operator F r so that B ( rL ) =Ω r ( L ) F r . Since Ω r ( L ) is an isometry, and B ( rL ) is a contraction, F ∗ r F r = B ( rL ) ∗ B ( rL ) < I ,so that F r is therefore also a contraction. Again using that Ω r ( L ) is an isometry, one can verifythat each component of F r commutes with the right shifts, so that F r = F r ( L ) is a left operator-valued multiplier [11, Theorem 1.2]. Moreover, F r = F r ( L ) has dense range by construction, andis therefore NC outer.) The goal of this section is to prove that B ( rL ) is injective for 0 < r ≤ d ∈ N and d = ∞ . In the notation of the previous discussion: Lemma 5.1. If Q r SOT → Q , then P r SOT → P .Proof. This is a consequence of the fact that Q r ⊗ I d SOT → Q ⊗ I d . The convergence is immediate,if d ∈ N . For d = ∞ , this is equivalent to Q r σ − SOT → Q . This latter claim follows from the fact thatthe Q r are bounded and [50, Lemma 2.5]. (cid:3) The main part of the following lemma is implicit in the work of Davidson and Pitts [11].
Lemma 5.2.
Let A ( L ) ∈ H ∞ ( B d N ) ⊗ L ( H , J ) be any left multiplier, and set M r := Ran ( A ( rL )) −k·k .The wandering dimension of M r is non-decreasing as r ↑ . Furthermore, if W r is the wanderingsubspace of M r and P r is the projection onto W r , then W r = ( P r Φ r W ) −k·k .Proof. It suffices to show that for every 0 < r <
1, we have that W r = P r Φ r ( W ) −k·k . Indeed, thisimplies that dim ( W r ) ≤ dim ( W ). Moreover, for 0 < t < r ≤
1, set C ( L ) = A ( rL ) and s = t/r ,then A ( tL ) = C ( sL ) and applying the lemma to C ( L ) will yield dim ( W t ) ≤ dim ( W r ).Now fix 0 < r <
1, and let W = { w , · · · , w k } be an orthonormal basis of W . Note that W r = Φ r W ⊂ M r . Moreover, note that Φ r ( M ) −k·k = M r since Φ r ( H ( B d N )) is dense in H ( B d N ).Since rR j ⊗ I K Φ r = Φ r R j ⊗ I K for every 1 ≤ j ≤ d and NC polynomials in R ⊗ I J acting on W generate a dense linear subspace of M , we conclude that W r is R ⊗ I J − cyclic in M r . Let P ⊥ r be,as above, the projection onto M r ⊖ W r . Let w ∈ W r ⊖ P r W r and u ∈ W r = Φ r W be arbitrary.Write u = P r u + P ⊥ r u . For every multi-index α we obtain that h w, R α ⊗ I J u i = h w, R α ⊗ I J P r u i = 0 . The first equality follows from the fact that M r ⊖ W r is R − invariant and the second since w iswandering and orthogonal to P r W r . Since W r is a R ⊗ I J − cyclic subset of M r , we conclude that w ≡ W r = W P r Φ r W = ( P r Φ r W ) −k·k . (cid:3) Let T r = I − Q r be the projection onto Ran ( B ( rL )) ⊥ , for r ∈ (0 , Proposition 5.3.
The projections T r SOT → T = I − Q = I − B ( L ) B ( L ) ∗ . Lemma 5.4.
For any < r ≤ , Sing( B ( rL )) is the set of all ( Z, y ) so that ( rZ, y ) ∈ Sing(Θ) . roof. One has y ∗ Θ( rZ ) = 0 if and only if ( rZ, y ) ∈ Sing(Θ) . (cid:3) Lemma 5.5.
Let S ⊂ H ( B d N ) be any linear subspace. A vector x ∈ H ( B d N ) is orthogonal to Φ r S if and only if x r = Φ r x is orthogonal to S .Proof. This follows immediately from the fact that Φ r is self-adjoint. (cid:3) Lemma 5.6.
Any NC Szeg¨o kernel vector, K { Z, y, v } for Z ∈ B dn , y, v ∈ C n , n ∈ N ∪ {∞} is givenby the formula: K { Z, y, v } = X α ∈ F d ( Z α v, y ) C n L α . For any r ∈ [0 , , Φ r K { Z, y, v } = K { rZ, y, v } .Proof. (of Proposition 5.3) We have thatRan ( T ) = Ran ( B ( L )) ⊥ = _ ( Z,y ) ∈ Sing( B ) K { Z, y, v } . Choosing a countable dense subset of kernel vectors and applying Gram-Schmidt orthgonalization(and using that linear combinations of NC kernels are NC kernels) we obtain an orthonormal basis { K { Z ( n ) , y n , v n }} ∞ n =1 , for Ran ( B ( L )) ⊥ . (Each ( Z ( n ) , y n ) belongs to Sing( B ), in fact, since linear combinations of NCkernels are NC kernels: K { Z, y, v } + cK { W, x, u } = K { Z ⊕ W, y ⊕ c · x, v ⊕ u } , and if ( Z, y ) , ( W, x ) ∈ Sing( B ), so is ( Z ⊕ W, y ⊕ c · x ) for any c ∈ C , see Proposition 3.8. This is,however, not germane for our arguments here.) Given any N ∈ N , and any 0 < r <
1, we define T r ( N ) as the orthogonal projection onto _ (cid:8) K { r − Z ( n ) , y n , v n } (cid:12)(cid:12) ≤ n ≤ N and k Z ( n ) k < r (cid:9) . Here, note that for any NC Szeg¨o kernel in the above set, k Z ( n ) k /r < H ( B d N ). If we choose 0 < R N < R N = max ≤ n ≤ N k Z ( n ) k , then for any r ∈ ( R N , T r ( N ) is the projection onto _ ≤ n ≤ N K { r − Z ( n ) , y n , v n } . We write T ( N ) := T ( N ). Since Φ r K { r − Z ( n ) , y n , v n } = K { Z ( n ) , y n , v n } ∈ Ran ( B ( L )) ⊥ byLemma 5.6, each of the K { r − Z ( n ) , y n , v n } belongs to Ran ( T r ) = Ran ( B ( rL )) ⊥ by Lemma 5.5.(In fact, ( r − Z ( n ) , y n ) ∈ Sing( B ( rL )) by Lemma 5.4.) It follows that T r ( N ) ≤ T r for any r ∈ (0 , K { r − Z ( n ) , y n , v n } −→ r ↑ K { Z ( n ) , y n , v n } , o that T r ( N ) SOT → T ( N ) as r ↑
1. Moreover it is clear that T ( N ) SOT → T .Consider the net ( T r ) r ∈ (0 , . This is a net of projections, and hence is uniformly bounded for0 < r ≤
1. Any subsequence ( T r k ), for which r k ↑ W OT convergent subsequence. Let ( T r k )be any such W OT − convergent subsequence so that as r k ↑ T r k W OT → e T . Then, for any N ∈ N , h x, e T x i = lim h x, T r k x i≥ lim h x, T r k ( N ) x i = h x, T ( N ) x i . Here, we note that since r k ↑
1, we have that eventually r k > R N . This proves that e T ≥ T ( N ) forany N ∈ N , and hence e T ≥ T . Further note that e T is positive semi-definite, and it is a contraction:Since T r k W OT → e T , h x, e T x i = lim k h x, T r k x i ≥ . Moreover, k e T x k = lim |h T r k x, e T x i|≤ lim sup k T r k x kk e T x k≤ k x kk e T x k . This proves that k e T x k ≤ k x k , and k e T k ≤
1. Let x = e T y be any vector in Ran (cid:16) e T (cid:17) . Then x k := T r k y w → x , where w denotes weak convergence. By Lemma 5.5, we know that for each k , x k ∈ Ran ( B ( r k L )) ⊥ , so that h k := Φ r k x k ∈ Ran ( B ( L )) ⊥ . Since each Φ r k is a contraction and sois e T , the sequence h k is uniformly bounded. Then, for any α ∈ F d ,lim k ( h k ) α = lim k r | α | k ( x k ) α = x α , since r k ↑
1, and ( x k ) α → x α since x k converges weakly to x . Since α ∈ F d is arbitrary and thesequence ( h k ) is uniformly bounded, it follows that h k w → x (converges weakly to x ). Moreover, each h k ∈ Ran ( B ( L )) ⊥ , and closed subspaces are weakly closed, so that x = wk − lim h k ∈ Ran ( B ( L )) ⊥ and we conclude that Ran (cid:16) e T (cid:17) ⊆ Ran ( T ). Since e T is a positive semi-definite contraction and T isa projection, T e T = e T , and T e T = e T = e T ∗ = e T T = T e T T ≤ T. This proves that e T ≤ T . Earlier we proved that e T ≥ T , and we conclude that T = e T = W OT − lim k T r k . Since the subsequence T r k was an arbitrary W OT − convergent subsequence so that r k ↑ T r converges in W OT to T as r ↑
1. Since each T r , T are projections,we then obtain that T r → T in the strong operator topology. (cid:3) emark 5.7. Since B ( rL ) converges SOT − ∗ to B ( L ) as r ↑ e.g. [23, Lemma 6.3]), itfollows that B ( rL ) B ( rL ) ∗ SOT → B ( L ) B ( L ) ∗ = Q as r ↑
1. Since Q is a non-trivial projection, itsspectrum is { , } , and it follows that for any t ∈ (0 , χ [0 ,t ] ( B ( rL ) B ( rL ) ∗ ) SOT → ( I − Q ) , and χ [ t, ( B ( rL ) B ( rL ) ∗ ) SOT → Q, where χ [ a,b ] denotes the characteristic function of the interval [ a, b ] [44, Theorem VIII.24 (b)]. Itdoes not immediately follow, however, that Q r = I − T r converges to Q because Q r = χ (0 , ( B ( rL ) ∗ B ( rL )) , and 0 belongs to the spectrum of Q , see [44]. The crucial fact that makes the above proof work isthat if B ( L ) is NC Blaschke, then Ran ( B ( L )) ⊥ is spanned by NC functions which are each analyticin an NC ball of radius greater than 1. Corollary 5.8. B ( rL ) is injective for r ∈ (0 , . Lemma 5.9. If = h ∈ Ker( B ( rL )) , there is an h ′ ∈ Ker( B ( rL )) so that h ′ (0) = 0 ∈ C N . If e = I − E is the NC idempotent so that Ran ( e ( rL )) = Ker( B ( rL )) , then e ∅ = e (0) ≡ vanishesidentically if and only if e ≡ is identically zero.Proof. Observe that Ker( B ( rL )) = Ker( B ( rL ) ∗ B ( rL )). Indeed if B ( rL ) h = 0 then B ( rL ) ∗ B ( rL ) h =0. Conversely, if B ( rL ) ∗ B ( rL ) h = 0, then0 = h h, B ( rL ) ∗ B ( rL ) h i = k B ( rL ) h k , and it follows that B ( rL ) h = 0.If h ∅ = h (0) = 0, Then h = R h = R h (1) + · · · + R d h ( d ) for some h ∈ H ( B d N ) ⊗ C N ⊗ C d . Then0 = R ∗ k B ( rL ) ∗ B ( rL ) h = B ( rL ) ∗ B ( rL ) h ( k ) , and it follows that h ( k ) ∈ Ker( B ( rL )) for every 1 ≤ k ≤ d . If h ( k ) ∅ = 0, then we can repeat thisprocess until we ultimately end up with a g ∈ Ker( B ( rL )) so that g (0) = 0. In more detail, if α ∈ F d is any word of minimal length so that h α = 0, then g := ( R α ) ∗ h ∈ Ker( B ( rL )), and g (0) = g ∅ = h α = 0.If e (0) ≡
0, then any h ∈ Ker( B ( rL )) = Ran ( e ( rL )) has the form h = e ( rL ) g for some g ∈ H ( B d N ) ⊗ C N , so that h (0) = e (0) g (0) = 0. Hence there is no h ∈ Ker( B ( rL )) so that h (0) = 0.If there was a non-zero h ∈ Ker( B ( rL )), then by the above argument there would be a non-zero g ∈ Ker( B ( rL )) so that g (0) = 0. We conclude that Ker( B ( rL )) = { } and e ≡ (cid:3) Proof. (of Corollary 5.8) We have proven that if Q r is the projection onto Ran ( B ( rL )) −k·k that Q r SOT → Q = B ( L ) B ( L ) ∗ . Consider the inner-outer factorization of B ( rL ) ∈ H ∞ ( B d N ) ⊗ C × N . Let { e k } Nk =1 be the standard orthonormal basis of C N . Then B k := B ( L )(1 ⊗ e k ) is an orthonormal basisfor the wandering space of Ran ( B ( L )). Let P r := Q r − R ( Q r ⊗ I d ) R ∗ , r ∈ (0 ,
1] be the orthogonal rojection onto the wandering subspace, W r , of Ran ( B ( rL )) −k·k . Then, by Lemma 5.1, P r SOT → P ,where P is the projection onto the wandering space of Ran ( B ( L )). Define ω r ; k := P r Φ r ( B k ), forevery 1 ≤ k ≤ N . Then each ω r ; k is a (potentially zero) wandering vector in Ran ( B ( rL )) −k·k , andsince P r SOT → P , Φ r SOT → I , and both nets are bounded, we have that ω r ; k = P r Φ r B k → P B k = B k ; 1 ≤ k ≤ N. (So for any fixed k , ω r ; k = 0 for r sufficiently close to 1.) Let N N := { , , · · · , N } , and set N N (0) := { j ∈ N N | ω r ; j = 0 } . We define a sequence of vectors in the wandering space of Ran ( B ( rL )) −k·k as follows: If k ∈ N N (0), so that ω r ; k = 0 we set Ω r ; k = 0. We then apply Gram-Schmidtorthogonalization to the (ordered) sequence:( ω r ; k ) k ∈ N N \ N N (0) . This produces an orthonormal sequence of vectors which we label in order by the elements of N \ N N (0). Combining this with the previous sequence of zero vectors indexed by N N (0) yieldsthe sequence (Ω r ; k ) Nk =1 , consisting of wandering vectors in W r so that the non-zero elements of thissequence form an orthonormal set. (And Ω r ; k = 0 if and only if k ∈ N N (0).) Note that for anyfixed k ∈ { , ..., N } , Ω r ; k converges to B k in Fock space norm as r ↑ k ∈ N N and r sufficiently close to 1, Ω r ; k = 0. Further observe, by Lemma 5.2, that the set, { ω r ; k } hasdense linear span in the wandering space of Ran ( B ( rL )) −k·k so that the set, { Ω r ; k } k ∈ N N \ N N (0) , is an orthonormal basis of wandering vectors for Ran ( B ( rL )) −k·k . The wandering dimension, N r ≤ N , of Ran ( B ( rL )) −k·k , is then the cardinality of the set N N \ N N (0). We then define: e Ω r ( L ) := (cid:16) M L Ω r ;1 , · · · , M L Ω r ; N (cid:17) : H ( B d N ) ⊗ C N → H ( B d N ) , and Ω r ( L ) := (Ω r ; j ( L )) j ∈ N N \ N N (0) . Observe that each non-zero Ω r ; j ( L ) = M L Ω r ; j (for j ∈ N N \ N N (0)), is an isometric, or inner leftmultiplier. It follows that e Ω r ( L ) ∈ H ∞ ( B d N ) ⊗ C × N is a partially isometric left multiplier andΩ r ( L ) ∈ H ∞ ( B d N ) ⊗ C × N r is the inner left multiplier obtained from e Ω r ( L ) by deleting any zeroentries. The inner-outer factorization of B ( rL ) is then B ( rL ) = Ω r ( L ) F r ( L ) , where F r ( L ) := Ω r ( L ) ∗ B ( rL ). If N r < N , we add a tail end of N − N r zeroes to Ω r ( L ) to obtainˆΩ r ( L ) := (Ω r ( L ) , , · · · , ∈ H ∞ ( B d N ) ⊗ C × N . If we set ˆ F r ( L ) := ˆΩ r ( L ) ∗ B ( rL ) then note that we still have B ( rL ) = ˆΩ r ( L ) ˆ F r ( L ) , here ˆ F r ( L ) is simply F r ( L ) with N − N r rows of zeroes added to make it ‘square’. In particular,since Ω r ( L ) is an isometry, we have thatKer( B ( rL )) = Ker( F r ( L )) = Ker( ˆ F r ( L )) . Observe that there is a unitary basis permutation matrix U r ∈ C N × N so that e Ω r ( L ) = ˆΩ r ( L )( I H ( B d N ) ⊗ U r ) . If for example, N = 3, N r = 2 and e Ω r = (Ω r ;1 , , Ω r ;3 ) , ˆΩ r = (Ω r ;1 , Ω r ;3 , , then, U r = , satisfies ˆΩ r ( L )( I H ( B d N ) ⊗ U r ) = e Ω r ( L ). If we then define, e F r ( L ) := e Ω r ( L ) ∗ B r ( L )= ( I H ( B d N ) ⊗ U ∗ r ) ˆΩ r ( L ) ∗ B ( rL )= ( I H ( B d N ) ⊗ U ∗ r ) ˆ F r ( L ) , we see that Ker( B ( rL )) = Ker( ˆ F r ( L )) = Ker( e F r ( L )) . We claim that e Ω r ( L ) converges in W OT to B ( L ). Indeed, each component Ω r ; k converges to B k = B k ( L )1 in Fock space norm, so that Ω r ; k ( Z ) → B k ( Z ) in the NC unit ball. This pointwiseconvergence and the uniform boundedness of the Ω r ; k ( L ) , B k ( L ) (these are all isometries or 0)implies W OT convergence of Ω r ; k ( L ) to B k ( L ) for any fixed k (see for example [47, Lemma 2.5]). Toprove that the entire row e Ω r ( L ) converges in W OT to B ( L ), let h ∈ H ( B d N ) ⊗ C N and g ∈ H ( B d N )be any fixed vectors. Given any ǫ > M ∈ N sufficiently large so that if h = h ...h N , then , N X M +1 k h k k H ( B d N ) < ǫ. Then, (cid:12)(cid:12)(cid:12) h ( e Ω r ( L ) − B ( L )) h , g i H ( B d N ) (cid:12)(cid:12)(cid:12) ≤ ǫ · k g k + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X k =1 h (Ω r,k ( L ) − B k ( L )) h k , g i H ( B d N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , which can be made arbitrarily small as r ↑ r ; k ( L ) converges in W OT to B k ( L ). ince the adjoint map is W OT − continuous, it then follows that e Ω r ( L ) ∗ W OT → B ( L ) ∗ . Finally,since B ( rL ) SOT → B ( L ), we obtain that e F r ( L ) = e Ω r ( L ) ∗ B ( rL ) W OT → I H ( B d N ) ⊗ I N . (Here, note that if A k and B k are uniformly bounded nets of operators on a Hilbert space so that A k W OT → A and B k SOT → B , then A k B k converges in the W OT to AB .) Since e F r ( L ) converges in W OT to I H ( B d N ) ⊗ I N , it follows that (cid:16) c , e F r (0) c ′ (cid:17) C N = h ⊗ c , e F r ( L )(1 ⊗ c ′ ) i→ (cid:0) c , c ′ (cid:1) C N , and this proves that e F r (0) ∈ C N × N converges in W OT to I N . As observed previously, Ker( B ( rL )) =Ker( e F r ( L )) so that B ( rL ) h = 0 implies e F r ( L ) h = 0. However, if e F r ( L ) = X α L α ⊗ e F r,α , and h = X β L β ⊗ h β ∈ Ker( B ( rL )) , then, 0 = e F r ( L ) h = X γ L γ ⊗ X α · β = γ e F r,α h β . All coefficients must vanish, so that in particular, e F r (0) h (0) = 0 . Now given any c , c ′ ∈ C N , we have that e ( rL )1 ⊗ c ∈ Ker( B ( rL )) = Ker( F r ( L )) = Ker( e F r ( L )) . It follows that 0 = e F r ( L ) e ( rL )1 ⊗ c , and in particular,0 = e F r (0) e (0) c . Then, 0 = (cid:16) c ′ , e F r (0) e (0) c (cid:17) C N → (cid:0) c ′ , e (0) c (cid:1) C N . Since c , c ′ ∈ C N were arbitrary we conclude that e (0) = 0. By Lemma 5.9, we conclude that e ≡ B ( rL ) is injective for 0 < r ≤ (cid:3) Theorem 5.10 (NC Blaschke-Singular-Outer factorization) . Any H ∈ H p ( B d N ) , p ∈ { , ∞} , has aunique Blaschke-Singular-Outer factorization: H = B · S · F ; B, S ∈ H ∞ ( B d N ) , F ∈ H p ( B d N ) , where B is an NC Blaschke inner, Sing( B ) = Sing( H ) , S is NC singular inner and F is an NCouter function. The factors are unique up to constants of unit modulus. roof. By the NC inner-outer factorization, any H ∈ H p ( B d N ), p = 2 or p = ∞ , factors as H = Θ · F for an NC inner Θ ∈ H ∞ ( B d N ) and an NC outer F ∈ H p ( B d N ) [11, Corollary 2.2, Corollary 2.3], [3,Theorem 2.1]. By Proposition 4.1 and Corollary 4.7,Θ = B · S = ( B , ··· , B N ) S ...S N ! , for a Blaschke row-inner B and a column-inner S , both of length N . In Corollary 4.8, we constructedan NC idempotent, e , e ( rL ) ∈ H ∞ ( B d N ) ⊗ C N × N for r ∈ [0 , B ( rL )) = Ran ( e ( rL )).Moreover, if E ( rL ) = I H ( B d N ) ⊗ I N − e ( rL ), then E ( rL ) = S ( rL ) H r ( L ), is an NC idempotent and H r ( L ) S ( rL ) = I H ( B d N ) , so that H r ( L ) ∈ H ∞ ( B d N ) ⊗ C × N is a left inverse for S ( rL ). (Also recallthat we can write H r ( L ) = H ( rL ) by Corollary 4.8 and Remark 4.10.) Corollary 5.8 shows that e ≡ S ( rL ) H ( rL ) = E ( rL ) = I H ( B d N ) ⊗ I N for any fixed 0 < r <
1. This means that thediagonal components obey: S k ( rL ) H k ( rL ) = I H ( B d N ) = H k ( rL ) S k ( rL ) . On the other hand, in Corollary 4.7 we proved that H ( rL ) is a left inverse for S ( rL ) so that I H ( B d N ) = H ( rL ) S ( rL ) = N X k =1 H k ( rL ) S k ( rL ) = N · I H ( B d N ) . This proves N = 1, and then S ( rL ) is an invertible left scalar multiplier with inverse H ( rL ). Inparticular, the NC variety of S is the empty set so that S ( S ) = H ( B d N ) and S is an NC singularinner function. (cid:3) When d = 1, we recover the classical Blaschke-Singular-Outer factorization with a new operator-theoretic proof: Corollary 5.11.
Given any h ∈ H p ( D ) , p ∈ { , ∞} , the NC Blaschke-Singular-Outer factorizationof h and the classical Blaschke-Singular-Outer factorization of h coincide. That is, if h = b · s · f is the classical Blaschke-Singular-Outer factorization of h , then the range of b ( M z ) = M b is thesingularity space of h .Proof. As observed in the introduction, if h = b · s · f is the classical Blaschke-Singular-Outerfactorization of h ∈ H p ( D ), p ∈ { , ∞} , thenRan ( M b ) = D ( h ) := (cid:26) f ∈ H (cid:12)(cid:12)(cid:12)(cid:12) fh ∈ Hol( D ) (cid:27) , is the set of all H functions which are divisible by h . On the other hand, if h = B · S · F is theNC Blaschke-Singular-Outer factorization of h obtained by setting d = 1 in Theorem 5.10 above,then it is clear from [11, Corollary 2.2] that F = f , and it remains to show thatRan ( M B ) = S ( h ) = { g ∈ H | y ∗ g ( Z ) = 0 ∀ ( Z, y ) ∈ Sing( h ) } , oincides with Ran ( M b ) = D ( h ). Clearly g ∈ D ( h ) if and only if every zero of h is a zero of g ofgreater or equal multiplicity. If w ∈ D is a zero of h of order n , consider W := (cid:18) w ǫ. .. .. . (cid:19) ∈ C ( n +1) × ( n +1) , where we choose 0 < ǫ < | w | so that W is a strict contraction. The image of W under h is h ( W ) = h ( w ) ǫh ′ ( w ) ǫ h ′ ( w )2! ··· ǫ n h ( n )( w ) n ! h ( w ) . ... .. ǫh ′ ( w ) h ( w ) , which vanishes identically as h has a zero of order n at w ∈ D . It follows that for any y ∈ C n +1 ,( W, y ) ∈ Sing( h ), so that any g ∈ S ( h ) is necessarily such that g ( W ) ≡
0. This is equivalentto w being a zero of g ∈ H of order at least n , and we conclude that S ( h ) ⊆ D ( h ) so thatRan ( M B ) ⊆ Ran ( M b ). Conversely, if ( Z, y ) ∈ Sing( h ) then,0 = y ∗ h ( Z )= y ∗ b ( Z ) s ( Z ) f ( Z ) , where s ( Z ) f ( Z ) is invertible, by spectral mapping, since s, f are non-vanishing in D . This provesthat y ∗ b ( Z ) = 0 for any ( Z, y ) ∈ Sing( h ) so that b = M b ∈ S ( h ) = Ran ( M B ), b = M B g = Bg ,for some g ∈ H . If p ∈ C [ z ] is any analytic polynomial, then M b p = M p b = M p M B g = M B pg ∈ Ran ( M B ) . Since M b C [ z ] is dense in Ran ( M b ), we conclude that Ran ( M b ) ⊆ Ran ( M B ) so that M b , M B havethe same range. Since b, B are inner functions in D with the same range, they are equal up to aunimodular constant. Without loss of generality B = b and F = f so that S = s as well. (cid:3) The infinite level.
A natural question is whether it is really necessary to include the infinitelevel, Sing ∞ ( H ) in our definition of NC variety. Our current operator-theoretic proof of the NCBlaschke-Singular Outer factorization theorem seems to rely on this. Namely, one can define the finite NC variety : Sing N ( H ) := G n ∈ N Sing n ( H ) , and the finite singularity space : S N ( H ) := { h ∈ H ( B d N ) | y ∗ h ( Z ) = 0 ∀ ( Z, y ) ∈ Sing N ( H ) } , and this is again a closed R − invariant subspace. Applying similar factorization arguments to thosein the proof of Proposition 4.1 to an NC inner H = Θ ∈ H ∞ ( B d N ) again yields:Θ( L ) = B ′ ( L ) S ′ ( L ) = ( B ′ ( L ) , ··· , B ′ N ( L ) ) S ′ ( L ) ...S ′ N ( L ) ! , or some ‘finite level’ NC Blaschke inner row, B ′ , i.e. Ran ( B ′ ( L )) = S N (Θ), and a ‘finite level’NC inner column, S ′ , where N ∈ N ∪ {∞} . If Θ( L ) = B ( L ) S ( L ) is the ‘infinite level’ (scalar)NC Blaschke-Singular factorization of Θ given by Theorem 5.10, it could be that Ran ( B ( L )) =Ran ( B ′ ( L )), so that B ′ ( L ) = B ( L ) up to a unimodular constant, and B ′ ( L ) is scalar. If this werethe case, unrestrictedly, then there would be no need to include the infinite level in our definition ofleft NC variety. While we currently do not know whether or not this is the case, we can show thatif p ∈ C { z , ..., z d } is any NC polynomial with NC Blaschke-Singular-Outer factorization p = BSF ,then B = B ′ is determined by the finite NC variety of p . Proposition 5.13. If p ∈ C { z , ..., z d } , then any ( Z, y ) ∈ Sing ∞ ( p ) can be approximated by finitedimensional ( Z ( k ) , y ( k ) ) ∈ Sing n k ( p ) , n k < ∞ , in the sense that K { Z, y, v } = wk − lim k →∞ K { Z ( k ) , y ( k ) , v } . In particular, S ( p ) = S N ( p ) . Proof.
Suppose that m is the homogeneous degree of p , and that ( Z, y ) ∈ Sing ∞ ( p ). Define thesubspace K := _ | α |≤ m ( Z α ) ∗ y ⊆ C ∞ := ℓ ( N ) , where we assume y ∈ C ∞ . Define the row contaction, X j := P K Z j | K , the compression of Z to the finite dimensional subspace K , and set x := y ∈ K . We claim that( X, x ) ∈ Sing N ( p ) where N := dim( K ). Indeed, this is easy to verify for m = 1. If m > X ∗ j X ∗ k x = P K Z ∗ j P K Z ∗ k x = P K Z ∗ j P K Z ∗ k y = P K Z ∗ j Z ∗ k y, and similarly, for any | α | ≤ m , ( X ∗ ) α x = P K ( Z ∗ ) α x = P K ( Z ∗ ) α y. It follows that p ( X ) ∗ x = P K p ( Z ) ∗ y = 0 , so that ( X, x ) ∈ Sing N ( p ). For any n ≥ m let K ( n ) := _ | α |≤ n ( Z α ) ∗ y, nd set X ( n ) j := P n Z j | K ( n ) , where P n := P K ( n ) . This produces a sequence of finite-dimensionalsingularity points ( X ( n ) , y ) ∈ Sing N n ( p ) , so that K { X ( n ) , y, v } w → K { Z, y, v } . Indeed, by Lemma 5.6, K { X ( n ) , y, v } = X α h X ( n ) α v, y i L α , where, for any fixed | α | < n , h X ( n ) α v, y i = h v, (( P n ZP n ) α ) ∗ y i = h Z α P n v, y i . For any fixed α ∈ F d , h Z α P n v, y i → h Z α v, y i , so that h X ( n ) α v, y i −→ n → ∞ h Z α v, y i . Since each k X ( n ) k ≤ k Z k <
1, the NC kernel vectors K { X ( n ) , y, v } are uniformly bounded inFock space norm. This, combined with the convergence of their coefficients implies that K { X ( n ) , y, v } converges weakly to K { Z, y, v } . In particular, S N ( p ) = _ ( Z,y ) ∈ Sing N ( p ) K { Z, y, v } = _ ( Z,y ) ∈ Sing( p ) K { Z, y, v } = S ( p ) . (cid:3) Another related and perhaps easier question is whether there exists an H ∈ H ∞ ( B d N ), such thatSing N ( H ) = ∅ , but Sing( H ) = ∅ ? A positive answer to this question, of course, implies that onecannot dispense with the infinite level. However, a negative answer does not tell us to what extentSing( H ) is determined by Sing N ( H ).6. NC Blaschke and Singular Examples
Homogeneous NC polynomials and NC Blaschke inners.
In this example we will showthat every homogeneous free polynomial p ∈ C { z , ..., z d } is a constant multiple of a Blaschke inner.Let p ∈ H ∞ ( B d N ) be a homogeneous polynomial. Since p ( L ) = M Lp is a constant times an isometry,we may assume without loss of generality that p ( L ) is an isometry, i.e. p is inner. It is immediatethat Sing( p ) is homogeneous in the first coordinate, i.e. , if ( Z, y ) ∈ Sing( p ), then for every λ ∈ D ,( λZ, y ) ∈ Sing( p ). Let f ∈ S ( p ) and ( Z, y ) ∈ Sing( p ). Write f = P ∞ n =0 f n , the Taylor-Taylorseries of f at 0 ∈ B d , where f n are the homogeneous components. Then we immediately have0 = y ∗ Z π e − inθ f ( e iθ Z ) dθ π = y ∗ f n ( Z ) . ence for every n ∈ N , f n ∈ S ( p ). By the Bergman Nullstelensatz [18, Theorem 6.3] we have that f n = pg , for some homogeneous g . This proves that f is in the range of p ( L ) and we conclude that S ( p ) = Ran ( p ( L )) so that p is Blaschke, by definition.6.2. The Weyl algebra relation.
For any w ∈ D , consider the M¨obius transformation: µ w ( z ) := z − w − wz . Lemma 6.3. If V ∈ L ( H ) is an isometry then µ w ( V ) is also an isometry.Proof. Consider: µ w ( V ) ∗ µ w ( V ) = ( I − wV ∗ ) − ( V ∗ − w )( V − w )( I − wV ) − . Expand the middle term: ( V ∗ − w )( V − w ) = I − wV − wV ∗ + | w | = ( I − wV ∗ )( I − wV ) , and this proves the claim. (cid:3) In the classical Hardy space literature, any M¨obius transformation composed with a contractiveanalytic function in the disk is sometimes called a
Frostman shift [14, 13], see also [15, Section 2.6].
Corollary 6.4. (NC inner Frostman shifts) If Θ ∈ H ∞ ( B d N ) is inner, then for any w ∈ D , Θ w := µ w (Θ) = ( I − w Θ) − (Θ − wI ) , is also inner. The main result of this subsection will be:
Theorem 6.5.
Let V ( Z ) be any inner NC homogeneous polynomial. For any w ∈ D , the NCFrostman shift V w ( Z ) = µ w ( V ( Z )) is Blaschke. Again, in the classical Hardy space literature, given any inner θ ∈ H ∞ , and any w ∈ D , thereis a natural unitary (isometric and onto) multiplier, C w ( z ), from ( θH ) ⊥ onto ( θ w H ) ⊥ , whereas before θ w = µ w ( θ ) is the w − Frostman shift of θ . The unitary multiplication operator, M C w :( θH ) ⊥ → ( θ w H ) ⊥ is sometimes called a Crofoot Transform [7], [15, Theorem 6.3.1].
Proposition 6.6. (NC Crofoot Transform) Left multiplication by C w ( Z ) := p − | w | ( I n − w Θ( Z )) − , is an isometry from (cid:0) Θ( L ) H ( B d N ) (cid:1) ⊥ onto (cid:0) Θ w ( L ) H ( B d N ) (cid:1) ⊥ . roof. The NC kernel for the orthogonal complement of Ran (Θ w ( L )) is K Θ w ( Z, W ) = K ( Z, W ) − Θ w ( Z ) K ( Z, W )Θ w ( W ) ∗ , = ( I − w Θ( Z )) − · (( I − w Θ( Z )) K ( Z, W )( I − w Θ( W ) ∗ ) − (Θ( Z ) − wI ) K ( Z, W )(Θ( W ) ∗ − wI )) | {z } =: G ( Z,W ) · ( I − w Θ( W ) ∗ ) − . The expression G ( Z, W ) can be expanded as: K ( Z, W ) − w Θ( Z ) K ( Z, W ) − wK ( Z, W )Θ( W ) ∗ + | w | Θ( Z ) K ( Z, W )Θ( W ) ∗ − Θ( Z ) K ( Z, W )Θ( W ) ∗ + wK ( Z, W )Θ( W ) ∗ + w Θ( Z ) K ( Z, W ) − | w | K ( Z, W )= (1 − | w | ) ( K ( Z, W ) − Θ( Z ) K ( Z, W )Θ( W ) ∗ )= (1 − | w | ) K Θ ( Z, W ) . Hence, K Θ w ( Z, W ) = (1 − | w | )( I − w Θ( Z )) − K Θ ( Z, W )( I − w Θ( W ) ∗ ) − , and the claim follows readily from this formula. (cid:3) Let V ∈ C { z , ..., z d } be an inner free homogeneous polynomial of degree n ∈ N , fix w ∈ D , andconsider the operator I − V w /n r L !! − = (cid:18) I − wr n V ( L ) (cid:19) − , where w /n is any n th root of w , and 0 < r < | w | r n < , ⇒ , i.e. | w | /n < r < , to ensure that this operator is well-defined as a convergent geometric series. Lemma 6.7.
Given any h ∈ Ran ( V ( L )) ⊥ = Ker( V ( L ) ∗ ) , and | w | /n < r < , h ( r ) := (cid:18) I − wr n V ( L ) (cid:19) − h ∈ Ker( V w ( rL ) ∗ ) . Proof.
Expand h ( r ) as a convergent geometric series and calculate: V ( rL ) ∗ h ( r ) = r n V ( L ) ∗ ∞ X k =0 w k r n · k V ( L ) k h = r n V ( L ) ∗ h | {z } =0 + r n ∞ X k =1 w k r n · k V ( L ) k − h = r n wr n ∞ X k =1 w k − r n · ( k − V ( L ) k − h = wh ( r ) . his proves that every h ( r ) is an eigenvector of V ( rL ) ∗ to eigenvalue w . It then follows that, V w ( rL ) ∗ h ( r ) = µ w ( V ( rL )) ∗ h ( r ) = ( I − wV ( rL )) − ( V ( rL ) ∗ − wI ) h ( r ) | {z } =0 . (cid:3) The above lemma implies that the following linear span of NC Szeg¨o kernels, K := _ | w | /n Given < r ≤ , the left multipliers H r ( L ) := H ( rL ) are uniformly bounded below(and hence have closed ranges) for r sufficiently close to .Proof. Each of the left multipliers H r ( L ) are injective. By the open mapping theorem, it followsthat H r ( L ) is bounded below if and only if it has closed range. In particular, by assumption wehave that H ( L ) is bounded below, by say δ > 0. Since we further assume that H is in the NC diskalgebra, H ( rL ) → H ( L ) in operator norm as r ↑ < R < r > R implies that k H ( rL ) − H ( L ) k < ǫ , where ǫ := δ/ 2. Hence, for any x ∈ H ( B d N ), k H ( rL ) x k ≥ k H ( L ) x k − k ( H ( L ) − H ( rL )) x k ≥ δ k x k , so that H ( rL ) is uniformly bounded below by δ/ R < r ≤ (cid:3) Proof. (of Theorem 6.10) To prove that Θ is Blaschke, we need to show that S (Θ) = Ran (Θ( L )).Since H has closed range, Ran ( H ( L )) = Ran (Θ( L )), where H ( L ) = Θ( L ) F ( L ) is the inner-outerfactorization of H . Hence, by Lemma 4.6, we need to show that S ( H ) = Ran ( H ( L )).Fix any 0 < r < 1, and consider any x ∈ Ran ( H ( rL )) ⊥ . Observe that the pair ( rL, x ) ∈ Sing ∞ ( H ). It follows that if g is any element in S ( H ), then h x, g ( rL )1 i H ( B d N ) = 0 , or any x ∈ Ran ( H ( rL )) ⊥ , and this proves that g r = g ( rL )1 ∈ Ran ( H ( rL )) −k·k , for any 0 < r < r sufficiently close to 1, g r = g ( rL )1 ∈ Ran ( H ( rL )) , since H ( rL ) has closed range for r sufficiently close to 1 by the previous lemma. In conclusion, g r = H ( rL ) x ( r ) , for some x ( r ) ∈ H ( B d N ). Observe that the net ( x ( r ) ) is uniformly bounded above (for r close to1). By the previous lemma, there is an ǫ > < R < r > R implies that H ( rL )is bounded below by ǫ . Hence, for such r , since the net ( g r ) is convergent and hence uniformlybounded in norm, k g r k = k H ( rL ) x ( r ) k≥ ǫ k x ( r ) k , proving that k x ( r ) k is uniformly bounded for r > R . By weak compactness, there is a weaklyconvergent subsequence x k := x ( r k ) , which therefore converges pointwise to some x ∈ H ( B d N ).Hence, for any Z ∈ B d N , g ( r k Z ) = H ( r k Z ) · x k ( Z ) ↓ ↓ ↓ g ( Z ) = H ( Z ) · x ( Z ) , so that g = H ( L ) x ∈ Ran ( H ( L )). This completes the proof. (cid:3) NC singular inner examples. If B ∈ [ H ∞ ( B d N )] , i.e. B belongs to the NC Schur class ofall contractive NC functions in B d N , then B ( L ) is a contraction on the NC Hardy space. By [49,Chapter 8], (provided B ( L ) = I H ( B d N ) ) B ( L ) is the co-generator of a C semigroup of contractionson H ( B d N ). Namely, if H B ( L ) := ( I − B ( L )) − ( I + B ( L )) , is the inverse Cayley transform of B , then H B ( L ) is a closed, densely-defined accretive operator(numerical range in the right half-plane), so that H B ( Z ) belongs to the NC Herglotz class of locallybounded (holomorphic) NC functions in B d N with positive semi-definite real part:Re ( H B ( Z )) ≥ n , Z ∈ B dn . Since 1 is not an eigenvalue of B ( L ), [49, Theorem III.8.1] implies that B t ( L ) := exp( − tH B ( L )); t ≥ , is a SOT − continuous one-parameter monoid of contractions on H ( B d N ), so that B t ( Z ) ∈ [ H ∞ ( B d N )] belongs to the NC Schur class for every t ≥ 0. Moreover, by [49, Proposition III.8.2], B t ( L ) will bean isometry on H ( B d N ) for every t ≥ i.e. B t will be NC inner, if and only if B ( L ) is NC inner.It further follows that if B ( L ) is NC inner, then every B t ( L ) will be an NC singular inner since B t ( Z ) = exp( − H B ( Z )) , Z ∈ B dn , s clearly pointwise invertible in B d ℵ . This provides a large class of examples of NC singular innerfunctions, and products of such NC singular inner functions are again NC singular inner. It isunclear whether or not all NC singular inners can be obtained in this way.7. Outlook The NC Blaschke-Singular-Outer factorization raises several natural questions. Classically, theinner factor of any polynomial in D is a finite Blaschke product, and hence a rational analyticfunction with poles outside of the open disk. Rational functions have been studied extensively inthe NC setting by several authors [55, 28, 56, 25, 43, 17]. Question 1. If p ∈ C { z , ..., z d } is any NC polynomial, is its NC inner factor Blaschke? Is it anNC rational function? Is the NC outer factor an NC polynomial?Frostman’s theorem states that given any inner function, θ , in the unit disk, ‘almost all’ ofits M¨obius transformations are Blaschke inner. There is also a theory of so-called indestructibleBlaschke products , these are Blaschke inner functions so that their images under any M¨obius trans-formation are again Blaschke products. In particular, the Blaschke inner factor of any polynomial(a finite Blaschke product) is indestructible [46], [15, Frostman’s Theorem, Theorem 2.6.1]. Question 2. Does an NC analogue of Frostman’s theorem hold? If the inner factor of any NCpolynomial is Blaschke, is it indestructible?Any Blaschke inner in the disk is a (potentially) infinite product of Blaschke factors : B w ( z ) := z − w − wz . Similarly one could define NC Blaschke factors as irreducible NC Blaschke inner functions, B , withthe property that there are no non-trivial NC Blaschke inners B , B so that B = B B . A finalquestion is whether there is a nice characterization of NC Blaschke factors. Appendix A. Idempotents in H ∞ ( B d N ) ⊗ C n × n Theorem A.1. Let E ∈ H ∞ ( B d N ) ⊗ C n × n be an idempotent, then there exists an orthogonal pro-jection P ∈ C n × n and an S ∈ GL n ( H ∞ ( B d N )) , such that E = S − (cid:16) I H ( B d N ) ⊗ P (cid:17) S. In particular, this implies that there are no non-trivial finitely generated projective modules over H ∞ ( B d N ) and thus H ∞ ( B d N ) is a semi-free ideal ring, see [6, Section 2.3]. Proof. Let M = Ran ( E ) = Ker( I − E ) and K = Ker( E ) and note that M + N = H ( B d N ) ⊗ C n . Inparticular, the Friedrichs angle between M and K is non-zero. Additionally, the spaces M and K are R ⊗ I n − invariant and closed. Let W M and W K be the wandering subspaces of M and K , respectively.Note that since H ( B d N ) ⊗ C n surjects onto M and K , that m = dim ( W ) M , k = dim ( W ) K ≤ n .(This follows as in the proof of Lemma 5.2.) Let V M ( L ) : F d ⊗ C m → H ( B d N ) ⊗ C n be the inner left ultiplier in H ∞ ( B d N ) ⊗ C n × m with image M and similarly V K ( L ) ∈ H ∞ ( B d N ) ⊗ C n × k be the isometricleft multiplier with image K . Consider S ( L ) ∈ H ∞ ( B d N ) ⊗ C n × ( k + m ) given by S = ( V M , V K ).Clearly, S is surjective and bounded. Furthermore, since M ∩ K = { } , S is also injective andthus has a bounded inverse. For every 1 ≤ i ≤ d , S ( R i ⊗ I n ) = ( R i ⊗ I m ) S so that S = S ( L ).Multiplying by S − on both left and right we get that ( R i ⊗ I n ) S − = S − ( R i ⊗ I m ). Thus S − ∈ H ∞ ( B d N ) ⊗ C ( k + m ) × n .Note that S ( L ) : H ( B d N ) ⊗ C k + m → H ( B d N ) ⊗ C n is surjective and thus m + k ≥ n . Similarly S − is surjective and thus n ≥ m + k . 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