aa r X i v : . [ m a t h . F A ] J un L ¨OWNER’S THEOREM IN SEVERAL VARIABLES
MIKL ´OS P ´ALFIA
Abstract.
In this paper we establish a multivariable non-commutativegeneralization of L¨owner’s classical theorem from 1934 characterizingoperator monotone functions as real functions admitting analytic con-tinuation mapping the upper complex half-plane into itself. The non-commutative several variable theorem proved here characterizes severalvariable operator monotone functions, not assumed to be free analyticor even continuous, as free functions that admit free analytic continua-tion mapping the upper operator poly-halfspace into the upper operatorhalfspace over an arbitrary Hilbert space. We establish a new abstractintegral formula for them using non-commutative topology, matrix con-vexity and LMIs. The formula represents operator monotone and op-erator concave free functions as a conditional expectation of a Schurcomplement of a linear matrix pencil on a tensor product operator alge-bra. This formula is new even in the one variable case. The results canbe applied to any of the various multivariable operator means that havebeen constructed in the last three decades or so, including the Karchermean. Thus we obtain an explicit, closed formula for these operatormeans of several positive operators. Introduction
In 1934 L¨owner proved his influential theorem on operator monotone func-tions of a real variable which states that all such functions are preciselygiven by holomorphic functions mapping the upper complex half-plane intoitself [41]. Many different proofs of this result appeared using different tech-niques [10, 30, 21, 22, 54, 60]. Some monographs comparing and givingdetails about the various techniques are [11, 16, 56]. Most of the proofs arebased on establishing an integral characterization formula for real functionsthat are operator monotone and then extending the domain of the functionto the upper complex half plane by analytic continuation through the for-mula. These results were used in 1980 by Kubo and Ando [32] to constructa two-variable theory of operator means, which in some sense provides acharacterization for two-variable operator monotone functions with specialproperties. The article [32] has proven to be an influential step towardsdeveloping a general theory of multivariable operator means, which are by
Date : June 19, 2018.2000
Mathematics Subject Classification.
Primary 46L52, 47A56 Secondary 47A64.
Key words and phrases. operator monotone function, free function, operator mean,Karcher mean. definition, uniquely induced by a certain class of normalized multivariableoperator monotone functions. In [32] two-variable operator monotone func-tions with some additional properties are considered and characterized. Inparticular the two-variable geometric mean has received a great deal of at-tention, many authors were considering various different methods to extendthe geometric mean to more than two non-commuting operator variables[6, 8, 9, 13, 15, 33, 36, 39, 43, 46, 47, 48]. The problem was also moti-vated by practical applications in medical imaging [8, 43], radar imagingtechnology [9] and many others. The various different techniques of exten-sions seemingly lead to different geometric means in more than two non-commuting variables, and in the last ten years, since the paper [6], in thematrix analysis community, that is the current state of the art. The so calledKarcher mean emerged among the different multivariable noncommutativegeometric means as the most important one due to its connections to metricgeometry as the center of mass, see for instance [39]. Calculation of thisparticular mean is of interest in the numerical linear algebra community, seefor instance [9, 15, 43] and many others. The best technique that we cur-rently have relies on approximations using gradient descent and Newton-likeiterations, see for example [40] and the references therein.On the other hand the theory of analytic functions in several non-commutingindeterminates were enjoying its own success, leading to recent representa-tion theorems for free analytic functions [23] with positive real part on thenon-commutative operatorial unit k -ball, see for example [52, 53, 58] amongthe numerous contributions to this field. Another recent setting is consid-ering free analytic functions mapping the upper operator poly-halfspace Π k into the upper operator halfspace Π := { X ∈ B ( E ) : ℑ X = i ( X − X ∗ ) > } where E is a Hilbert space and k is a positive integer [50, 2]. In this field op-erator monotonicity has been studied only recently in the case of commutingtuples of operators [3] and the focus remained restricted to free functionsthat are a priori assumed to be free analytic. In [3] an operator modeltheoretic representation formula is proved for k -variable continuously differ-entiable real functions that are operator monotone for commuting tuples ofmatrices. For a very recent contribution characterizing free analytic matrixmonotone functions by establishing an operator model theoretic representa-tion formula, we refer to [50] using techniques from [3] related to operatormonotonicity and the so called Herglotz representation formula for free an-alytic functions with positive real part on the non-commutative operatorial k -ball [52, 53]. The obtained representations in [50] based on [52] rely onthe theory of noncommutative disc algebras developed for non-commutativepower series representations and the Cuntz-Toeplitz algebra generated bypartial isometries induced by the left- and right creation operators on thefull Fock space. The free analytic functions are then represented by theslice map, or in other words, the conditional expectation of the inverse ofcertain linear combinations of tensor products on the Cuntz-Toeplitz alge-bra determined by a completely positive linear map which is itself uniquely ¨OWNER’S THEOREM IN SEVERAL VARIABLES 3 determined by the values of the analytic function, for more technical detailssee [52, 53]. There is also an earlier related result [24], where it is provedthat a free rational, hence automatically analytic, k -variable matrix convexfunction has a representation formula superficially similar to the ones in[3, 50]. The representation formula called the butterfly representation pro-vided in [24] for rational free analytic functions is the closest relative of ourrepresentation formula obtained in this paper.Let P n denote the cone of positive definite n -by- n matrices, which is asubset of the cone of positive definite operators denoted by P ( E ) over the(infinite dimensional) Hilbert space E . In this paper we adopt the conventionthat if X ∈ P ( E ) then this means that X is also lower bounded, henceinvertible, the same applies to the notation X >
0. We will denote by ˆ P thecone of positive semi-definite operators and similarly ˆ P n denotes its finite n -by- n -dimensional counterpart. Let S ( E ) denote the vector space of boundedself-adjoint operators over E and S n its finite dimensional counterpart whichis a subspace of S . Sometimes we will denote S ( E ) simply by S if certainassertion holds for elements of S ( E ) for any Hilbert space E . We will adoptthe following shorthand notation for k -tuples of operators in the above sets: X := ( X , . . . , X k )where k is a fixed positive integer. We also use the notation I := ( I, . . . , I )where I is the identity operator. The upper operator poly-halfspace is de-noted by Π k := { X ∈ B ( E ) k : ℑ X i > , ≤ i ≤ k } where the imaginary part of a bounded linear operator A is defined as ℑ A := A − A ∗ i . A several variable function F : D S ( E ) for a domain D ⊆ S ( E ) k definedfor all Hilbert spaces E is called a free or noncommutative function (NCfunction) if for all A, B ∈ S ( E ) k which are in the domain of F we have(1) F ( U ∗ A U, . . . , U ∗ A k U ) = U ∗ F ( A , . . . , A k ) U for all U − = U ∗ ∈B ( E ),(2) F (cid:18)(cid:20) A B (cid:21) , . . . , (cid:20) A k B k (cid:21)(cid:19) = (cid:20) F ( A , . . . , A k ) 00 F ( B , . . . , B k ) (cid:21) ,Operator monotonicity and concavity is defined in Definition 2.2 and Defini-tion 2.3 below. It must be noted that the author does not know any operatormean in the literature which is operator monotone and does not satisfy theproperties of free functions. In particular all the different geometric andoperator means discussed in [6, 13, 15, 33, 36, 39, 43, 46, 47, 48] satisfy thefree function property and of course monotonicity.The main result of this paper, the representation formula equivalent tooperator concavity, monotonicity and free analytic continuation to the upperoperator poly-halfspace Π k mapping it to the upper operator halfspace Πfor a free function F is as follows: MIKL ´OS P ´ALFIA
Theorem A.
Let E be a Hilbert space and let F : P ( E ) k P ( E ) be a freefunction. Then the following are equivalent: (a) F is operator monotone; (b) F is operator concave; (c) There exists a Hilbert space K , a closed subspace K ≤ K and thecorresponding orthogonal projection P K with range K , B i ∈ ˆ P ( K ) , ≤ i ≤ k with B ≥ P ki =1 B i and a state w ∈ B +1 ( K ) ∗ such that F ( X ) is the w -conditional expectation of the Schur complement ofthe linear pencil L B ( X ) = B ⊗ I + P ki =1 B i ⊗ X i pivoting on thesubspace K for all X ∈ P ( E ) k . I.e. for all X ∈ P ( E ) k we have F ( X ) = w ( B , ) ⊗ I + k X i =1 w ( B i, ) ⊗ ( X i − I ) − ( w ⊗ I ) (" B , ⊗ I + k X i =1 B i, ⊗ ( X i − I ) B , ⊗ I + k X i =1 B i, ⊗ ( X i − I ) − " B , ⊗ I + k X i =1 B i, ⊗ ( X i − I ) where B i, ( A, v ) := P K B i P K ,B i, ( A, v ) := P K B i ( I − P K ) ,B i, ( A, v ) :=( I − P K ) B i P K ,B i, ( A, v ) :=( I − P K ) B i ( I − P K );(d) F has a free analytic continuation to Π k , mapping Π k to Π . In Theorem A we restricted the domain and range of the free functions,but in principle M¨obius transformations can be used to transform the domainof operator monotone functions to P and the range assumption in Theorem Ais basically equivalent to the assumption that the function is bounded frombelow. The exact method to use M¨obius transformations on the domainof F is discussed in the last section of the paper. Also due to the freeanalytic continuation part of Theorem A the techniques in [50] apply, toobtain a general formula using the Cayley transform to connect Π k andthe noncommutative operatorial k -ball and transform the noncommutativeHerglotz formula established in [53] for free holomorphic functions with realpart on the noncommutative operatorial k -ball. This in principle follows thetechnique used in [11] to establish a Nevanlinna type formula for holomorphicfunctions mapping the upper complex half-plane into itself. ¨OWNER’S THEOREM IN SEVERAL VARIABLES 5 In order to prove the Theorem A, we first establish the equivalence ofoperator monotonicity and operator concavity over certain domains. Theproof of the equivalence between operator monotonicity and concavity in theone variable case has already appeared as early as in [22, 21]. In particularthe argument given in [21] goes through in several variables as well withminor modifications. Then by concavity we deduce the norm continuityof operator monotone functions. Then we introduce the hypograph of freefunctions as the saturation of their graphs:hypo( F ) = (hypo( F )( E )) := ( { ( Y, X ) ∈ S ( E ) × P ( E ) k : Y ≤ F ( X ) } ) . Theorem 3.2 states that a free function F is operator concave/operatormonotone if and only if hypo( F ) is a matrix convex set in the sense ofWittstock. Then in Proposition 3.8 for each boundary point of hypo( F )( E )we construct a linear matrix pencil, similarly as in [18, 26] for other matrixconvex sets, that is positive semi-definite on the matrix convex set hypo( F )and it is singular at the boundary point. The proof of Proposition 3.8 inprinciple is similar to the one given in section 6 of [26], but it also worksfor complex Hilbert spaces and it is formulated in such a way, that it canbe applied directly in the infinite dimensional case as well, although for ourpurpose the finite dimensional version suffices.The next step is Theorem 4.4 which is an explicit extremal linear matrixinequality (LMI) solution formula based on the Schur complement that pro-vides a key reconstruction formula for the actual values F ( X ) v at arbitrarytuples X applied to arbitrary unit vectors v ∈ E . Then we construct a non-separable Hilbert space where all possible combinations of the points of thedomain of the free function F are listed as a direct summand giving a blockdiagonal operator acting on this space. Then we apply Theorem 4.4 to re-construct the value of the function at this block diagonal operator acting onvectors of this non-separable Hilbert space to get Lemma 4.5 and then applya compactness argument to obtain the representation formula Theorem 4.6.The compactness argument is inspired by the one given in [24] for provinglinear dependence of free rational expressions over finite dimensional spaces.Theorem 4.6 expresses operator concave/operator monotone functions as aconditional expectation of the Schur complement of a linear matrix pencilestablishing (c) in Theorem A. Then in the last section of the paper weestablish the free analytic continuation to obtain (d) in Theorem A. Herewe use Proposition 5.3, a version of a result appeared in [17] for matrices,providing an estimate on the norm of the Schur complement of a sectorialoperator.The approach outlined above provides a new proof of L¨owner’s theorem inthe one variable case as well. The representation formula based on the Schurcomplement is also new even in this case. We believe that this approach inthe one variable case has the advantage over the existing ones in that thereis no need to establish continuous differentiability of operator monotonefunctions by mollifier smoothing techniques. Also the linear matrix pencil MIKL ´OS P ´ALFIA that appears in (c) in Theorem A can be thought of as the direct sum of theLMI representations of the supporting linear functionals of the set hypo( F ),so the formula in (c) is intuitive from the convex geometrical point of view.2. Noncommutative functions, monotonicity and concavity
We put the following plausible assumptions on our operator valued severalvariable functions. Let E denote an arbitrary Hilbert space. Definition 2.1 (NC function, [58]) . A several variable function F : D ( E ) S ( E ) for a domain D ( E ) ⊆ S ( E ) k defined for all Hilbert spaces E is calleda free or noncommutative function (NC function) if for all E and all A, B ∈ D ( E ) ⊆ S ( E ) k (1) F ( U ∗ A U, . . . , U ∗ A k U ) = U ∗ F ( A , . . . , A k ) U for all U − = U ∗ ∈B ( E ),(2) F (cid:18)(cid:20) A B (cid:21) , . . . , (cid:20) A k B k (cid:21)(cid:19) = (cid:20) F ( A , . . . , A k ) 00 F ( B , . . . , B k ) (cid:21) .Note that this already includes the closure of the domain D ( E ) under di-rect sums and element-wise unitary conjugation. A free function F can beregarded as a graded function F : D ( E ) S ( E ) between the collection ofdomains D = ( D ( E )) and its range included in S ( E ) for each Hilbert space E .The second property means that NC functions respect direct sum decom-positions, while the first property is invariancy under unitary conjugations.Both assumptions are plausible, since in one variables, the functional cal-culus has these properties. Also see [3, 50, 53] which adopt the same as-sumptions, under the additional assumption of free analyticity. Moreovermany such functions considered as means of operators have these properties[6, 11, 12, 13, 15, 35, 37, 39, 43].The set S is equipped with a partial order, the positive definite order ≤ ,which means that for A, B ∈ S we have A ≤ B if and only if 0 ≤ B − A ,that is h ( B − A ) x, x i ≥ x ∈ E . For k -tuples A, B ∈ S k wedefine similarly(1) A ≤ B iff A i ≤ B i for all i = 1 , . . . , k. Definition 2.2 (Monotonicity) . An NC function F : P k S is said to beoperator monotone if whenever A ≤ B for A, B ∈ P k , we have F ( A ) ≤ F ( B ) . If this property is satisfied only in finite dimensions, then we say that theNC function F : P kn S n is n -monotone. Definition 2.3 (Concavity & Convexity) . A function F : P k S is said tobe operator concave if for all A, B ∈ P k and λ ∈ [0 , − λ ) F ( A ) + λF ( B ) ≤ F ((1 − λ ) A + λB ) ¨OWNER’S THEOREM IN SEVERAL VARIABLES 7 Similarly if this property is satisfied only for n -by- n matrices, then we saythat the NC function F : P kn S n is n -concave. Operator and n -convexity isdefined accordingly. Many times later on we consider concavity or convexityon subsets of P k .All the above definitions can be considered on other convex domains (or-der intervals), not just P k . Remark 2.1.
Notice that if a function F is n -monotone or n -concave, thenby the direct sum property it is m -monotone and m -concave accordinglyfor all 1 ≤ m ≤ n . Also if F is operator monotone or concave then it is n -monotone or n -concave for all finite n ≥ Lemma 2.1.
Let F be a concave function into S on an open convex set U in a normed linear space. If F is bounded from below in a neighborhood ofone point of U , then F is locally bounded on U .Proof. Suppose that F is bounded from below by M I for some M ∈ R on anopen ball B ( a, r ) with radius r around a . Let x ∈ U and choose ρ > z := a + ρ ( x − a ) ∈ U . If λ = 1 /ρ , then V = { v : v = (1 − λ ) y + λz, y ∈ B ( a, r ) } is a neighborhood of x = (1 − λ ) a + λz , with radius (1 − λ ) r .Moreover, for v ∈ V we have F ( v ) ≥ (1 − λ ) F ( y ) + λF ( z ) ≥ (1 − λ ) M I + λF ( z ) ≥ KI for some K ∈ R . To show that F is bounded above in the same neighbor-hood, choose arbitrarily v ∈ V and notice that 2 x − v ∈ V . By concavity F ( x ) ≥ F ( v ) / F (2 x − v ) /
2, which yields F ( v ) ≤ F ( x ) − F (2 x − v ) ≤ F ( x ) − KI. (cid:3)
We equip B ( E ) k and similarly S k and ˆ P k with the norm k X k := k X i =1 k X i k for a tuple X ∈ B ( E ) k . Proposition 2.2 (see also Proposition 3.5.4 in [45]) . An operator concavefunction F : P k S which is locally bounded from below, is continuous inthe norm topology.Proof. Let U ⊆ P k be an open norm bounded neighborhood with respectto the operator norm k · k . Let A ∈ U and r > B ( A, r ) := { X ∈ U : k X − A k < r } ⊆ U . Let X, Y ∈ B ( A, r ) and X = Y such that α := k Y − X k < r . Let(2) Z := Y + rα ( Y − X ) . MIKL ´OS P ´ALFIA
Then k Z − A k ≤ k Y − A k + rα k Y − X k < r, i.e. Z ∈ B ( A, r ). By (2) we have Y = rr + α X + αr + α Z, so by operator concavity of F we get F ( Y ) ≥ rr + α F ( X ) + αr + α F ( Z ) , which after rearranging yields F ( X ) − F ( Y ) ≤ αr + α ( F ( X ) − F ( Z )) ≤ αr + α M I ≤ αr M I, where the real number
M > F on U in theform of − M I ≤ F ( X ) − F ( Z ) ≤ M I by Lemma 2.1. Now exchange therole of X and Y in the above to obtain the reverse inequality F ( Y ) − F ( X ) ≤ αr M I. From the above pair of inequalities we get k F ( Y ) − F ( X ) k ≤ Mr k Y − X k proving continuity. (cid:3) A net of operators { A i } i ∈I is called increasing if A i ≥ A j for i ≥ j and i, j ∈ I . Also { A i } i ∈I is bounded from above if there exists some realconstant K > A i ≤ KI for all i ∈ I . It is well known thatany bounded from above increasing net of operators { A i } i ∈I has a leastupper bound sup i ∈I A i such that B j := A j − sup i ∈I A i converges to 0 in thestrong operator topology. Similarly if we have a decreasing net of boundedoperators that is bounded from below, then the net converges to its greatestlower bound.The next characterization result is an extension of Theorem 2.1 in [21]to several variables. The proof is analogous to that of Theorem 2.1. Weconsider the finite dimensional situation, but its proof is presented in sucha way that it works also in the infinite dimensional setting as well. Proposition 2.3.
Let F : P k n S n be a n -monotone function. Then itsrestriction F : P kn S n is n -concave, moreover it is norm continuous.Proof. Let
A, B ∈ P kn and let λ ∈ [0 , n -by-2 n blockmatrix is unitary V := (cid:20) λ / I n − (1 − λ ) / I n (1 − λ ) / I n λ / I n (cid:21) . ¨OWNER’S THEOREM IN SEVERAL VARIABLES 9 Elementary calculation reveals that V ∗ (cid:20) A B (cid:21) V = (cid:20) λA + (1 − λ ) B λ / (1 − λ ) / ( B − A ) λ / (1 − λ ) / ( B − A ) (1 − λ ) A + λB (cid:21) . Set D := − λ / (1 − λ ) / ( B − A ) and notice that for any given ǫ > (cid:20) λA + (1 − λ ) B + ǫI
00 2 Z (cid:21) − V ∗ (cid:20) A B (cid:21) V ≥ (cid:20) ǫI DD Z (cid:21) if Z ≥ (1 − λ ) A + λB . The last k -tuple of block matrices is positive semi-definite if Z ≥ D i /ǫ for all 1 ≤ i ≤ k . So, for sufficiently large positivedefinite Z we have V ∗ (cid:20) A B (cid:21) V ≤ (cid:20) λA + (1 − λ ) B + ǫI
00 2 Z (cid:21) . For such
Z >
0, by the 2 n -monotonicity of F we get F (cid:18) V ∗ (cid:20) A B (cid:21) V (cid:19) ≤ (cid:20) F ( λA + (1 − λ ) B + ǫI ) 00 F (2 Z ) (cid:21) . We also have that F (cid:18) V ∗ (cid:20) A B (cid:21) V (cid:19) = V ∗ (cid:20) F ( A ) 00 F ( B ) (cid:21) V = (cid:20) λF ( A ) + (1 − λ ) F ( B ) λ / (1 − λ ) / ( F ( B ) − F ( A )) λ / (1 − λ ) / ( F ( B ) − F ( A )) (1 − λ ) F ( A ) + λF ( B ) (cid:21) , hence we obtain that(3) λF ( A ) + (1 − λ ) F ( B ) ≤ F ( λA + (1 − λ ) B + ǫI ) . Now since F is 2 n -monotone, F ( X + ǫI ) for ǫ > F ( X ), thus the right strong limit F + ( X ) := inf ǫ> F ( X + ǫI ) = lim ǫ → F ( X + ǫI )exists for all X ∈ P k n . Hence for any ǫ >
0, using (3), we obtain λF + ( A )+(1 − λ ) F + ( B ) ≤ λF ( A + ǫI )+(1 − λ ) F ( B + ǫI ) ≤ F ( λA +(1 − λ ) B +2 ǫI ) . Taking the strong limit ǫ →
0+ we obtain that λF + ( A ) + (1 − λ ) F + ( B ) ≤ F + ( λA + (1 − λ ) B ) , i.e. the NC function F + is n -concave. Also F ( X ) ≤ F + ( X ) ≤ F ( X + ǫI )for all ǫ >
0, since F is monotone increasing, hence F + is bounded frombelow on order bounded sets, so by Proposition 2.2 F + is norm continuouson order bounded sets, since every point A ∈ S has a basis of neighborhoodsin the norm topology that are order bounded sets. As the last step, againby the monotonicity of F we have F + ( X − ǫI ) ≤ F ( X ) ≤ F + ( X ) , and since F + is norm continuous we get F = F + by taking the norm limit ǫ → ǫ →
0+ in (3) and concludethat F is n -concave and continuous. (cid:3) Since the above proof also works in infinite dimensions we have the fol-lowing result.
Corollary 2.4.
An operator monotone NC function F : P k S is operatorconcave and norm continuous. The reverse implication is also true if F is bounded from below: Theorem 2.5.
Let F : P k ˆ P be operator concave ( n -concave) NC func-tion. Then F is operator monotone ( n -monotone).Proof. The proof goes along the lines of Theorem 2.3 in [21]. (cid:3)
By the above if we wish to characterize operator monotone functions F : P k P , then it suffices to characterize operator concave ones.We close the section with a semi-continuity property for operator mono-tone NC functions. Proposition 2.6.
Let F : P k ˆ P be an operator monotone NC func-tion. Then for any bounded from above increasing net of k -tuple of opera-tors { A i } i ∈I with A i ∈ P k we have that F is strongly upper semi-continuousalong increasing nets, i.e. sup i ∈I F ( A i ) ≤ F (cid:18) sup i ∈I A i (cid:19) . Proof.
We have that A j ≤ sup i ∈I A i component-wise for all j ∈ I , i.e.( A j ) l ≤ sup i ∈I ( A i ) l for all 1 ≤ l ≤ k and j ∈ I . Thus by the monotonicityof F we have F ( A j ) ≤ F (cid:18) sup i ∈I A i (cid:19) for all j ∈ I , so it follows thatsup j ∈I F ( A j ) ≤ F (cid:18) sup i ∈I A i (cid:19) . (cid:3) Supporting linear pencils and hypographs
In this section we will use the theory of matrix convex sets introducedfirst by Wittstock. Some references on free convexity and matrix convexsets are [18, 24, 25, 26, 27]. Let Lat( E ) denote the lattice of subspaces of E . The notation K ≤ E means that K is a closed subspace of E , hence aHilbert space itself. Definition 3.1 (Matrix/Freely convex set) . A graded collection C = ( C ( K )),where each C ( K ) ⊆ S ( K ) k and K is a Hilbert space, is a bounded τ open/closed matrix convex or freely convex set if ¨OWNER’S THEOREM IN SEVERAL VARIABLES 11 (i) each C ( K ) is open/closed in the τ topology;(ii) C respects direct sums, i.e. if ( X , . . . , X k ) ∈ C ( K ) and ( Y , . . . , Y k ) ∈ C ( N ) and Z j := (cid:20) X j Y j (cid:21) for Hilbert spaces K, N , then ( Z , . . . , Z k ) ∈ C ( K ⊕ N );(iii) C respects conjugation with isometries, i.e. if Y ∈ C ( K ) and V : N K is an isometry for Hilbert spaces K, N , then V ∗ Y V =( V ∗ Y V, . . . , V ∗ Y k V ) ∈ C ( N );(iv) each C ( K ) is bounded.The above definition has some equivalent characterizations under slightadditional assumptions. Definition 3.2.
A graded collection C = ( C ( K )), where each C ( K ) ⊆ S ( K ) k , is closed with respect to reducing subspaces if for any tuple of oper-ators ( X , . . . , X k ) ∈ C ( K ) and any corresponding mutually invariant sub-space N ⊆ K , we have that the restricted tuple ( ˆ X , . . . , ˆ X k ) ∈ C ( N ), whereeach ˆ X i is the restriction of X i to the invariant subspace N for all 1 ≤ i ≤ k . Lemma 3.1 (Lemma 2.3 in [27], § . Suppose that the graded col-lection C = ( C ( K )) , where each C ( K ) ⊆ S ( K ) k respects direct sums in thesense as in (ii) in Definition 3.1 and it respects unitary conjugation in thesense as in (iii) in Definition 3.1 with N = K . (1) If C is closed with respect to reducing subspaces then C is matrixconvex if and only if each C ( K ) is convex in the classical sense oftaking scalar convex combinations. (2) If C is (nonempty and) matrix convex, then , . . . , ∈ C (1) ifand only if C is closed with respect to simultaneous conjugation bycontractions. Given a set A ⊆ S we define its saturation assat( A ) := { X ∈ S : ∃ Y ∈ A, Y ≥ X } . Similarly for a graded collection C = ( C ( K )), where each C ( K ) ⊆ S ( K ), its saturation sat( C ) is the disjoint union of sat( C ( K )) for each Hilbert space K . Definition 3.3 (Hypographs) . Let F : P k S be an NC function. Thenfor a fixed constant c >
0, we define its hypograph hypo( F ) as the gradedcollection of the saturation of its image, i.e.hypo( F ) = (hypo( F )( K )) := ( { ( Y, X ) ∈ S ( K ) × P ( K ) k : Y ≤ F ( X ) } ) , Theorem 3.2.
Let F : P k S be an NC function. Then its hypograph hypo( F ) is a matrix convex set if and only if F is operator concave.Proof. Suppose first that F is operator concave. We will prove the matrixconvexity of hypo( F ) by establishing the properties in (1) of Lemma 3.1.By the definition of operator concavity and the convexity of P and the order intervals, it follows easily that for each Hilbert space K , hypo( F )( K ) isconvex in the usual sense of taking scalar convex combinations. To seethat hypo( F ) is closed with respect to reducing subspaces, assume that( Y, X ) ∈ hypo( F )( L ) with ( Y, X ) = ( ˆ
Y , ˆ X ) ⊕ ( Y , X ) and ( ˆ
Y , ˆ X ) ∈ S ( K ) × P ( K ) k , ( Y , X ) ∈ S ( N ) × P ( N ) k for Hilbert spaces K ⊕ N = L . Then since F is an NC function, it respects direct sums, hence Y ≤ F ( X ) = F ( ˆ X ) ⊕ F ( X ).Again by the definition of NC functions, we have F ( ˆ X ) ∈ S ( K ) and F ( X ) ∈ S ( N ). Since Y = ˆ Y ⊕ Y , it follows that ˆ Y ≤ F ( ˆ X ) and Y ≤ F ( X ), i.e.( ˆ Y , ˆ X ) ∈ hypo( F )( K ) and ( Y , X ) ∈ hypo( F )( N ).For the converse, suppose that hypo( F ) is a matrix convex set. First of allnotice that hypo( F ) is closed with respect to reducing subspaces. Indeed,similarly to the above assume that ( Y, X ) ∈ hypo( F )( L ) with ( Y, X ) =( ˆ
Y , ˆ X ) ⊕ ( Y , X ) and ( ˆ
Y , ˆ X ) ∈ S ( K ) × P ( K ) k , ( Y , X ) ∈ S ( N ) × P ( N ) k forHilbert spaces K ⊕ N = L . Then since F is an NC function, it respectsdirect sums, hence Y ≤ F ( X ) = F ( ˆ X ) ⊕ F ( X ). Again by the definition ofNC functions, we have F ( ˆ X ) ∈ S ( K ) and F ( X ) ∈ S ( N ). Since Y = ˆ Y ⊕ Y ,it follows that ˆ Y ≤ F ( ˆ X ) and Y ≤ F ( X ), i.e. ( ˆ Y , ˆ X ) ∈ hypo( F )( K )and ( Y , X ) ∈ hypo( F )( N ). So, again by (1) of Lemma 3.1 it follows thatfor each Hilbert space L , hypo( F )( L ) is convex in the usual sense. Noticealso that for each L we can recover the values of the NC function F , sincefor a fixed X ∈ P ( L ) k we have that F ( X ) = sup { Y ∈ S ( L ) : ( Y, X ) ∈ hypo( F )( L ) } . In other words for all t ∈ [0 ,
1] and
A, B ∈ P ( L ) k we havethat the tuple ( Y, X ) := (1 − t )( F ( A ) , A ) + t ( F ( B ) , B ) is in hypo( F )( L ),moreover since F ( X ) = sup { Y ∈ S ( L ) : ( Y, X ) ∈ hypo( F )( L ) } we have that(1 − t ) F ( A ) + tF ( B ) ≤ F ( X ) = F ((1 − t ) A + tB ) for all A, B ∈ P ( L ) k andHilbert space L , hence F is operator concave. (cid:3) The above Theorem 3.2 combined with Theorem 2.5 leads to the following:
Corollary 3.3.
Let F : P k ˆ P be an NC function. Then its hypograph hypo( F ) is a matrix convex set if and only if F is operator monotone. A sharpening of Theorem 3.2 is possible if we establish further continu-ity properties of NC functions. We will use the terminology of measurabledomain , compact domain and measurable operator function , continuous op-erator function given in [31] in the proof of the following auxiliary result. Lemma 3.4.
Suppose that F : P ( E ) k S ( E ) is a norm continuous NCfunction for any separable Hilbert space E . Then F is continuous in thestrong operator topology on bounded sets { X ∈ P ( E ) k : cI ≤ X i ≤ CI } forfixed constants < c < C .Proof. The set D := { X ∈ P ( E ) k : cI ≤ X i ≤ CI } for fixed constants 0 In [31] the above used theorems are proved for separableHilbert spaces, since the constructions use sets like the set of all Hilbertspaces, which does not exist, if we also include non-separable spaces. Henceone must restrict to separable Hilbert spaces (which as a matter of fact areisomorphic to each other for a fixed dimension) to obtain the results in [31].Given a Hilbert space E , the space of bounded linear operators B ( E ) asa von Neumann algebra has a unique predual B ( E ) ∗ , the Banach space oftrace-class operators. The topology induced by the duality ( B ( E ) , B ( E ) ∗ )is the σ - or ultra-weak operator topology. In other words this topology isgenerated by the closed subspace B ( E ) ∗ of normal linear functionals of thedual space B ( E ) ∗ . Theorem 3.5. Let E be separable and let F : P k P be an NC function.Then its restricted hypograph (4)hypo c,C ( F )( E ) := ( { ( Y, X ) ∈ S ( E ) ×{ X ∈ P ( E ) k : cI ≤ X i ≤ CI } : Y ≤ F ( X ) } ) for any given C > c > , is a σ -strongly/ σ -weakly closed matrix convex setif and only if F is operator concave if and only if F is operator monotone.Proof. By Theorem 3.2 and Corollary 3.3 F is operator concave if and onlyif F is operator monotone, if and only if hypo( F )( E ) is a matrix convexset. Then also by Corollary 2.4 F is norm continuous, hence by Lemma 3.4 F is strong operator continuous on norm bounded sets, and also { X ∈ P ( E ) k : cI ≤ X i ≤ CI } is a strong operator closed convex set. Hencehypo c,C ( F )( E ) is also strong operator closed and convex for each separable E . Since hypo c,C ( F )( E ) is a strong operator closed convex set, its closure inthe σ -weak operator topology is itself, since the weak and strong operatorclosure of convex sets of operators are the same and the weak and σ -weakoperator topologies coincide on norm bounded sets, see Theorem II.2.6 in[57]. (cid:3) In the case of non-separable E , we cannot use Lemma 3.4 to prove strongcontinuity of norm continuous functions on norm bounded sets. Instead we will consider some additional plausible assumption on F in the case ofnon-separable E . Assumption 1. Let F : P k S be an operator monotone NC function. If E is non-separable we assume that for any bounded from above increasingnet of k -tuple of operators { A i } i ∈I with A i ∈ P ( E ) k we have that (5) sup i ∈I F ( A i ) ≥ F (cid:18) sup i ∈I A i (cid:19) . For example the Karcher mean of positive definite operators satisfies theabove assumptions, see [34], as well as any Kubo-Ando mean [32] or theoperator means in [47].In what follows we will consider supporting linear pencils for a matrixconvex set, that are in one to one correspondence with supporting linearfunctionals coming from the Hahn-Banach Theorem given for topologicalvector spaces. Definition 3.4 (free ǫ -neighborhood) . Given ǫ > free ǫ -neighborhoodof 0, denoted by N ǫ , is for each Hilbert space K , the graded collection( N ǫ ( K )) where N ǫ ( K ) := { X ∈ S ( K ) k : k X j =1 k X j k < ǫ } . Definition 3.5 (linear pencil) . A linear pencil is an expression of the form L ( x ) := A + A x + · · · + A k x k where each A i ∈ S ( K ) for some Hilbert space K whose dimension is the size of the pencil L . The pencil is monic if A = I and then L is a monic linearpencil . We extend the evaluation of L from scalars to operators by tensormultiplication. In particular L evaluates at a tuple X ∈ S ( N ) k as L ( X ) := A ⊗ I + A ⊗ X + · · · + A k ⊗ X k . We then regard L ( X ) as a self-adjoint element of S ( K ⊗ N ).Let B +1 ( K ) ∗ ⊂ B ( K ) ∗ denote the convex set of positive semi-definiteoperators over K of trace one and let B +1 ( K ) ∗ denote the state space ofthe C ∗ -algebra B ( K ). Note that positive linear functionals on unital C ∗ -algebras attain their norm at the unit, hence B +1 ( K ) ∗ is convex and weak- ∗ compact by Banach-Alaoglu. Each element T ∈ B +1 ( K ) ∗ corresponds to astate on S ( K ) by X tr( XT )for X ∈ S ( K ). We will need a version of Proposition 6.4 in [26], beforegiving our version we state the following auxiliary result from [26]. Weendow B +1 ( K ) ∗ with the relative weak ∗ -topology induced by the duality( B h ( K ) , B h ( K ) ∗ ), where the superscript h denotes the self-adjoint part. ¨OWNER’S THEOREM IN SEVERAL VARIABLES 15 Lemma 3.6. Suppose F is a convex set of weak- ∗ continuous affine linearmappings f : B +1 ( K ) ∗ R with respect to a duality. If for each f ∈ F thereis a T ∈ B +1 ( K ) ∗ such that f ( T ) ≥ , then there is a T ∈ B +1 ( K ) ∗ such that f ( T ) ≥ for every f ∈ F .Proof. For f ∈ F , let B f := { T ∈ B +1 ( K ) ∗ : f ( T ) ≥ } ⊂ B +1 ( K ) ∗ . By hypothesis, each B f is nonempty and it suffices to prove that ∩ f ∈F B f = ∅ . Since each B f is compact, it suffices to prove that the collection { B f : f ∈ F } has the finite intersection property. Let f , . . . , f m ∈ F be given. Supposethat(6) ∩ mj =1 B f n = ∅ . Define F : B +1 ( K ) ∗ R m by F ( T ) := ( f ( T ) , . . . , f m ( T )) . Then F ( B +1 ( K ) ∗ ) is both convex and compact because B +1 ( K ) ∗ is and each f j , hence F , is weak- ∗ continuous affine linear. Moreover F ( B +1 ( K ) ∗ ) doesnot intersect R m + = { x = ( x , . . . , x m ) : x j ≥ j } . Hence by the Hahn-Banach theorem there exists a linear functional λ : R R such that λ ( F ( B +1 ( K ) ∗ )) < λ ( R m + ) ≥ 0. We can write λ as λ ( x ) = P mi =1 λ j x j . Since λ ( R m + ) ≥ 0, it follows that each λ j ≥ 0. We have λ j = 0 for at least one 1 ≤ j ≤ m , so without loss of generality we canassume that P mj =1 λ j = 1. Let f := m X j =1 λ j f j . Since F is convex, we have f ∈ F . On the other hand, f ( T ) = λ ( F ( T )),hence if T ∈ B +1 ( K ) ∗ , then f ( T ) < 0. Thus, for this f there does not exista T ∈ B +1 ( K ) ∗ such that F ( T ) ≥ 0, contradicting (6). (cid:3) Lemma 3.7. Let C = ( C ( K )) be a matrix convex set, where C ( K ) ⊆ S ( K ) k and (0 , . . . , ∈ C ( C ) . Let a linear functional Λ : S ( N ) k R be given fora fixed N with dim( N ) < ∞ . If Λ( X ) ≤ for each X ∈ C ( N ) , then thereexists a T ∈ B +1 ( N ) ∗ such that for each Hilbert space K , and each Y ∈ C ( K ) and each V : N K contraction V we have Λ( V ∗ Y V ) ≤ tr( V T V ∗ ) . Proof. Since dim( N ) < ∞ it follows that B +1 ( N ) ∗ = B +1 ( N ) ∗ . For a Hilbertspace K , a tuple Y ∈ C ( K ) and a V : N K contraction, define f Y,V : B +1 ( N ) ∗ R by f Y,V ( T ) := tr( V T V ∗ ) − Λ( V ∗ Y V ) . We claim that the collection F := { f Y,V : Y, V } is a convex set. Let λ i ≥ ≤ i ≤ n for a fixed integer n and let P ni =1 λ i = 1. Also let ( Y i , V i )be given where Y i ∈ C ( K i ) for a Hilbert space K i and V i : N K i bea contraction for each 1 ≤ i ≤ n . Let Z := ⊕ ni =1 Y i and let F denote thecolumn operator matrix with entries √ λ i V i . Then Z ∈ C ( ⊕ K i ) and F ∗ F = n X i =1 λ i V ∗ i V i ≤ n X i =1 λ i I = I. By definition n X i =1 λ i V ∗ i Y i V i = F ∗ ZF and n X i =1 λ i tr( V i T V ∗ i ) = tr( F T F ∗ )for T ∈ B +1 ( N ) ∗ . Hence n X i =1 λ i f Y i ,V i ( T ) = f Z,F ( T ) . If V has operator norm 1, we can choose a pure state γ ∗ ( · ) γ where γ is aunit vector in N such that1 = k V γ k = γ ∗ V ∗ V γ = tr( γγ ∗ V ∗ V ) = tr( V γγ ∗ V ∗ ) . Then for T = γγ ∗ it follows that f Y,V ( T ) = tr( V T V ∗ ) − Λ( V ∗ Y V ) = 1 − Λ( V ∗ Y V ) . Since V ∗ Y V ∈ C ( N ), the right hand side above is nonnegative. If thecontraction V does not have norm one, we can rescale it to have norm 1and follow the same argument to show that f Y,V ( T ) ≥ 0. So, for each f Y,V there exists a T ∈ B +1 ( N ) ∗ such that f Y,V ( T ) ≥ 0, moreover each f Y,V isweak- ∗ continuous. From Lemma 3.6 there exists a T ∈ B +1 ( N ) ∗ such that f Y,V ( T ) ≥ Y and V . (cid:3) For an arbitrary set S of a topological vector space we will denote by S ◦ its interior which is the union of all open sets contained in S . Below is amodified and generalized version of Proposition 6.4 in [26]. A similar resultis Theorem 5.4 in [18]. ¨OWNER’S THEOREM IN SEVERAL VARIABLES 17 Proposition 3.8. Let F : P k P be an operator monotone function andlet N be a fixed Hilbert space with dim( N ) < ∞ . Then for each A ∈ P ( N ) k and each unit vector v ∈ N there exists a linear pencil L F,A,v ( Y, X ) := B ( F, A, v ) ⊗ I − vv ∗ ⊗ Y + k X i =1 B ( F, A, v ) i ⊗ ( X i − I ) of size dim( N ) which satisfies the following properties: (a) B ( F, A, v ) i ∈ B + ( N ) ∗ and P ki =1 B ( F, A, v ) i ≤ B ( F, A, v ) ; (b) For all ( Y, X ) ∈ hypo( F ) we have L F,A,v ( Y, X ) ≥ ; (c) If c I ≤ A i ≤ c I for all ≤ i ≤ k and some fixed real constants c > c > , then tr { B ( F, A, v ) } ≤ F ( c ,...,c )min(1 ,c ) .Proof. By Theorem 3.2 hypo( F ) is a matrix convex set. Consider the trans-lated set H ( K ) := { ( Y, X ) ∈ S ( K ) × S ( K ) k : ( Y, ( X + I, . . . , X k + I )) ∈ hypo( F )( K ) } . Still H = ( H ( K )) is a matrix convex set. Moreover since F is positive we have F ( sI, . . . , sI ) = cI for some arbitrary small, but fixed s > X ≥ ( sI, . . . , sI ) we have F ( X ) ≥ cI by operator mono-tonicity. Hence hypo( F ) contains a free ǫ -neighborhood of 1, so H containsa free ǫ -neighborhood of 0 for small enough ǫ > 0. Then ( F ( A ) , A ) is inthe boundary of hypo( F )( N ), hence ( F ( A ) , ( A − I, . . . , A k − I )) is in theboundary of H ( N ). Consider the real valued function h : [ P ( N ) − I ] k R defined by h ( X ) := v ∗ F ( X + I ) v . Since F is an operator concave functionit follows that h is concave and by Corollary 2.4 it is also continuous in thenorm topology. It follows from the supporting hyperplane version of theHahn-Banach theorem for the finite dimensional vector space R × S ( N ) k ,more precisely Theorem 7.12 and 7.16 [4] that the norm continuous convexfunction g ( X ) := − h ( X ) = − v ∗ F ( X + I ) v has a subgradient at each interiorpoint of its domain, hence at ( A − I ) for A ∈ P ( N ) k . I.e. there exists anorm continuous linear functional λ in the dual space of S ( N ) k such that g ( X ) ≥ g ( A − I ) + λ ( X − A + I )for all X ∈ [ P ( N ) − I ] k . Hence it follows that there exists c ∈ R and l i inthe dual space of S ( N ) equipped with the norm topology such that(7) 1 ≥ c v ∗ F ( X ) v − k X i =1 l i ( X i − I )for all X ∈ P ( N ) k and(8) 1 = 1 c v ∗ F ( A ) v − k X i =1 l i ( A i − I ) . Without loss of generality we can assume that c > 0, hence it follows from(7) and the definition of hypo( F )( N ) that(9) 1 ≥ c v ∗ Y v − k X i =1 l i ( X i − I )for all ( Y, X ) ∈ hypo( F )( N ). From Lemma 3.7 and (9) there exists a T ∈B +1 ( N ) ∗ such that for each Hilbert space K , and each ( Y, X ) ∈ hypo( F )( K )and each V : N K contraction V we have(10) tr( V T V ∗ ) − c v ∗ V ∗ Y V v + k X i =1 l i ( V ∗ ( X i − I ) V ) ≥ . Since the dual space of S ( N ) (equipped with the norm topology) is the spaceof self-adjoint trace-class operators over N we have l i ( Z ) = tr { B i Z } for all Z ∈ S ( N ) where B i ∈ B h ( N ) ∗ is a trace class operator. Hence we can write(10) as(11) tr( V T V ∗ ) − c tr { vv ∗ V ∗ Y V } + k X i =1 tr { B i ( V ∗ ( X i − I ) V ) } ≥ , moreover (8) becomes(12) tr( T ) + k X i =1 tr { B i ( A i − I ) } = 1 c tr { vv ∗ F ( A ) } . Let L B,v denote the linear pencil L B,v ( Y, X ) := vv ∗ ⊗ Y − k X i =1 cB i ⊗ ( X i − I ) . Let K be a Hilbert space, let { e i } i ∈I denote an orthonormal basis of K andlet ( Y, X ) ∈ hypo( F )( K ). Then for an arbitrary unit vector γ = P j ∈I γ ∗ j ⊗ e j ∈ N ∗ ⊗ K we have γ ∗ L B,v ( Y, X ) γ = X i,j ∈I γ ∗ j vv ∗ γ i e ∗ i Y e j − k X l =1 γ ∗ j cB l γ i e ∗ i ( X l − I ) e j = X i,j ∈I tr ( vv ∗ γ i e ∗ i Y e j γ ∗ j − k X l =1 cB l γ i e ∗ i ( X l − I ) e j γ ∗ j ) = tr ( vv ∗ Γ ∗ Y Γ − k X l =1 cB l Γ ∗ ( X l − I )Γ ) ¨OWNER’S THEOREM IN SEVERAL VARIABLES 19 where Γ : N K is the contraction defined as Γ := P i ∈I e i γ ∗ i whereconvergence is in the ultraweak operator topology. Using (10) we havetr ( vv ∗ Γ ∗ Y Γ − k X l =1 cB l Γ ∗ ( X l − I )Γ ) ≤ tr(Γ cT Γ ∗ )= tr X i,j ∈I e j γ ∗ j cT γ i e ∗ i = X i,j ∈I γ ∗ j cT γ i e ∗ i e j = γ ∗ ( cT ⊗ I ) γ. Thus the linear pencil cT − L B,v defined by [ cT − L B,v ]( Y, X ) = cT ⊗ I − vv ∗ ⊗ Y + P ki =1 cB i ⊗ ( X i − I ) satisfies[ cT − L B,v ]( Y, X ) ≥ K ≤ E and ( Y, X ) ∈ hypo( F )( K ).Also computing as above (12) becomes(13) tr { E ([ cT − L B,v ]( F ( A ) , A )) } = 0with E = P i,j ∈I ( e ∗ i ⊗ e i )( e ∗ j ⊗ e j ) ∗ .Let B ( F, A, v ) := cT and B ( F, A, v ) i := cB i . Then (b) of the asser-tion is satisfied. Now we turn to the proof of (a). Observe that the point( − dI, ( I, . . . , I )) is in hypo( F ) for any scalar d ≥ F is positive, so wemust have L F,A,v ( − dI, ( I, . . . , I )) ≥ d ≥ 0. In other words wehave B ( F, A, v ) ⊗ I ≥ vv ∗ ⊗ − dI for all scalar d ≥ 0. This is only possibleif c ≥ ≤ i ≤ k with Y = 0, X j = I for j = i , X i = dI ; the point ( Y, X ) is in hypo( F )for any scalar d > 1, so we must have L F,A,v ( Y, X ) ≥ B ( F, A, v ) ⊗ I + B ( F, A, v ) i ⊗ dI ≥ d > B ( F, A, v ) i ≥ 0. Similarly the point (0 , ( dI, . . . , dI ))is in hypo( F ) for any scalar d > 0, hence similar consideration reveals that B ( F, A, v ) ⊗ I − (1 − d ) P ki =1 B ( F, A, v ) i ⊗ I ≥ d > d → 0+ we get P ki =1 B ( F, A, v ) i ≤ B ( F, A, v ) ,finishing the proof of property (a).It follows from (13) that tr { E L B,A,v ( F ( A ) , A ) } = 0 or equivalently(14) X i,j ∈I ( e i ⊗ e ∗ i ) L B,A,v ( F ( A ) , A )( e ∗ j ⊗ e j ) = 0 . Now assume that c I ≤ A i ≤ c I for all 1 ≤ i ≤ k and some fixed realconstants c > c > k X i =1 tr { B i ( A i − I ) } ≥ ( c − k X i =1 tr( B i ) and 1 = tr( T ) ≥ P ki =1 tr( B i ) ≥ 0. Since A i ≤ c I for all 1 ≤ i ≤ k , by theoperator monotonicity of F we also have v ∗ F ( A ) v ≤ v ∗ F ( c , . . . , c ) Iv = F ( c , . . . , c ), hence using (12) we get c = v ∗ F ( A ) v P ki =1 tr { B i ( A i − I ) } ≤ F ( c , . . . , c )1 + ( c − P ki =1 tr( B i ) ≤ F ( c , . . . , c )min(1 , c )which together with tr( B ( F, A, v ) ) = c tr( T ) proves (c). (cid:3) Explicit LMI solution formula Proposition 3.8 provides us a tool to find sufficiently many supportinglinear pencils of hypographs of our functions so that we can reconstruct thevalues of the functions at each point. We will need the following result from[7]. Theorem 4.1 (Theorem 3 cf. [7]) . Let A be a positive semi-definite lin-ear operator on a Hilbert space and S a subspace. Let the matrix of A be partitioned as A = (cid:20) A A A A (cid:21) with A : S S , A : S S ⊥ .Then ran( A ) ⊂ ran( A ) / and there exists a bounded linear operator C : S S ⊥ such that A = ( A ) / C and A = (cid:20) A − C ∗ C 00 0 (cid:21) + (cid:20) C ∗ A ) / (cid:21) (cid:20) C ( A ) / (cid:21) . The bounded positive semi-definite operator S ( A ) = A − C ∗ C is called theshorted operator or Schur complement of A . It satisfies S ( A ) ≤ A and it ismaximal among all self-adjoint operators X : S S such that X ≤ A . Note that the proof of the above result in [7] is based on the followingwell known result. Lemma 4.2 (Douglas’ lemma, cf. [20]) . Let A and B be bounded linear op-erators on a Hilbert space H . Then the following statements are equivalent: (1) ran( A ) ⊂ ran( B ) . (2) AA ∗ ≤ λ BB ∗ for some λ ≥ . (3) There exists a bounded linear operator C such that A = BC . Remark 4.1. Note that in the proof of the Douglas’ lemma the followingconstruction is used to prove (1) = ⇒ (3). The operator C is defined as C = B − A , where B is the restriction of B to the orthogonal complementof its kernel N ( B ) ⊥ , so that B − : ran( B ) 7→ N ( B ) ⊥ is a closed linearoperator, hence also C is a closed linear operator from H into N ( B ) ⊥ .From the closed graph theorem it follows that C is bounded.We will also use the following basic fact from time to time. Lemma 4.3. Let A ≥ be a positive semi-definite operator on some Hilbertspace H . If v ∗ Av = 0 for some v ∈ H , then Av = 0 . ¨OWNER’S THEOREM IN SEVERAL VARIABLES 21 Proof. Since A ≥ 0, it has a unique positive semi-definite square root A / .Hence 0 = v ∗ Av = k A / v k = 0 , so A / v = 0 and it follows that Av = 0. (cid:3) Theorem 4.4. Let F : P k P be an operator monotone function. Thenfor each A ∈ P ( N ) k with dim( N ) < ∞ and each unit vector v ∈ N we have F ( A ) v = v ∗ B , ( A, v ) v ⊗ Iv + k X i =1 v ∗ B i, ( A, v ) v ⊗ ( A i − I ) v − ( ( v ∗ ⊗ I ) " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) × " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) − × " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) ( v ⊗ I ) ) v (15) and (" B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) − " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) ( v ∗ ⊗ v )= X j ∈I " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) ( e ∗ j ⊗ e j ) , (16) where { e j } j ∈J is an orthonormal basis of N and B i, ( A, v ) := vv ∗ B i ( A, v ) vv ∗ ,B i, ( A, v ) := vv ∗ B i ( A, v )( I − vv ∗ ) ,B i, ( A, v ) :=( I − vv ∗ ) B i ( A, v ) vv ∗ ,B i, ( A, v ) :=( I − vv ∗ ) B i ( A, v )( I − vv ∗ )(17) for all ≤ i ≤ k and B i ( A, v ) = B ( F, A, v ) i .Moreover if c I ≤ A i ≤ c I for all ≤ i ≤ k and some fixed real constants c > c > , then (18) tr { B ( A, v ) } ≤ F ( c , . . . , c )min(1 , c ) . Proof. From Proposition 3.8 we have that(19) vv ∗ ⊗ F ( A ) ≤ B ( F, A, v ) ⊗ I + k X i =1 B ( F, A, v ) i ⊗ ( X i − I )and also by (14) X i,j ∈I ( e i ⊗ e ∗ i ) vv ∗ ⊗ F ( A )( e ∗ j ⊗ e j ) = X i,j ∈I ( e i ⊗ e ∗ i ) [ B ( F, A, v ) ⊗ I + k X i =1 B ( F, A, v ) i ⊗ ( X i − I ) ( e ∗ j ⊗ e j ) . (20)By (a) of Proposition 3.8 we can apply the Schur complement Theorem 4.1to (19), pivoting on the subspace v ∗ ⊗ N of N ∗ ⊗ N to get vv ∗ ⊗ F ( A ) ≤ B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) − " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) × " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) − × " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) (21)where the coefficients B i,xy ( A, v ) are defined by (17) and B i ( A, v ) := B ( F, A, v ) i .The inversion in 21 is justified by (a) in Proposition 3.8 alone or by the rangeinclusion result in Theorem 4.1 and that the operators have finite rank. No-tice that( vv ∗ ⊗ I ) X i,j ∈I ( e ∗ j ⊗ e j )( e i ⊗ e ∗ i ) ( vv ∗ ⊗ I ) = X i,j ∈I ( e ∗ j vv ∗ ⊗ e j )( vv ∗ e i ⊗ e ∗ i )= X i,j ∈I ( v ∗ ⊗ e j e ∗ j v )( v ⊗ v ∗ e i e ∗ i ) , =( v ∗ ⊗ v )( v ⊗ v ∗ ) ¨OWNER’S THEOREM IN SEVERAL VARIABLES 23 hence from (19), (20) and (21) we get v ∗ F ( A ) v = v ∗ B , ( A, v ) v ⊗ v ∗ Iv + k X i =1 v ∗ B i, ( A, v ) v ⊗ v ∗ ( A i − I ) v − ( v ⊗ v ∗ ) " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) × " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) − × " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) ( v ∗ ⊗ v ) + e (22)where e := ( ( v ⊗ v ∗ ) " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) × " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) − / + X j ∈I ( e j ⊗ e ∗ j ) − ( v ⊗ v ∗ ) " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) / " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) − / × " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) ( v ∗ ⊗ v )+ " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) / X j ∈I ( e ∗ j ⊗ e j ) − ( v ∗ ⊗ v ) . Notice that e is of the form e = x ∗ x , where x = D − / " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) ( v ∗ ⊗ v )+ D / X j ∈I ( e ∗ j ⊗ e j ) − ( v ∗ ⊗ v ) ,D = " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) . (23)It follows that e ≥ 0. The linear map l : B ( N ∗ ) ⊗ B ( N ) C ⊗ B ( N ) ∼ = B ( N )defined on simple tensors as l ( X ⊗ Y ) := v ∗ Xv ⊗ Y is (completely) positive,hence order preserving. So applying l to (21) we get F ( A ) ≤ v ∗ B , ( A, v ) v ⊗ I + k X i =1 v ∗ B i, ( A, v ) v ⊗ ( A i − I ) − ( v ∗ ⊗ I ) " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) × " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) − × " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) ( v ⊗ I ) . (24)Now since we have (24) and also (22) with e ≥ 0, we must have e = 0 in(22) which yields x = 0 in (23) hence (16) and also (15).Property (c) in Proposition 3.8 yields tr { B ( A, v ) } ≤ F ( c ,...,c )min(1 ,c ) . (cid:3) Definition 4.1 (Natural map) . A graded map F : S ( K ) k × K K foreach Hilbert space K is called a natural map if it preserves direct sums, i.e. F ( X ⊕ Y, v ⊕ w ) = F ( X, v ) ⊕ F ( Y, w )for X ∈ S ( K ) k , v ∈ K and Y ∈ S ( K ) k , w ∈ K , for Hilbert spaces K , K .For an NC function F : S k S we define the natural map F : S ( K ) k × K K for any Hilbert space K by F ( X, v ) := F ( X ) v for X ∈ S ( K ) k and v ∈ K . Indeed F is natural since F ( X ⊕ Y, v ⊕ w ) = F ( X ) v ⊕ F ( Y ) w = F ( X, v ) ⊕ F ( Y, w ) ¨OWNER’S THEOREM IN SEVERAL VARIABLES 25 for X ∈ S ( K ) k , v ∈ K and Y ∈ S ( K ) k , w ∈ K for Hilbert spaces K , K .Also notice that the function F ( X ) := v ∗ B , v ⊗ I + k X i =1 v ∗ B i, v ⊗ ( X i − I ) − ( v ∗ ⊗ I ) " B , ⊗ I + k X i =1 B i, ⊗ ( X i − I ) × " B , ⊗ I + k X i =1 B i, ⊗ ( X i − I ) − × " B , ⊗ I + k X i =1 B i, ⊗ ( X i − I ) ( v ⊗ I )for fixed v ∈ E and linear operators B i,xy such that the products aboveare well defined is an NC function, for example like in (15) of Theorem 4.4.Hence it defines a natural map in the same way by F ( X, w ) := F ( X ) w for X ∈ P ( K ) k and w ∈ K .Let S ( E ) := { v ∈ E : k v k = 1 } denote the unit sphere of the Hilbertspace E . In what follows, we will construct an auxiliary Hilbert space onwhich we will apply Theorem 4.4.For fixed real constants c > c > 0, let P c ,c ( E ) := { X ∈ P ( E ) : c I ≤ X ≤ c I } , Ω c ,c := P c ,c ( E ) k × S ( E )and let H := M dim( E ) < ∞ M ω ∈ Ω c ,c E. We equip H with the inner product x ∗ y := X dim( E ) < ∞ X ω ∈ Ω c ,c x ( ω ) ∗ y ( ω )for x, y ∈ H and we denote again by H the Hilbert space completion withrespect to this inner product. Definition 4.2. Let F : P k P be an operator monotone function. NowletΨ F ( X ) := M dim( E ) < ∞ M ( A,v ) ∈ Ω c ,c ( B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( X i − I ) − " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( X i − I ) × " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( X i − I ) − × " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( X i − I ) , where the coefficients B i,xy ( A, v ) for 0 ≤ i ≤ k and x, y ∈ { , } are as in(15) of Theorem 4.4. Then Ψ F ( X ) is a linear operator on H ⊗ E which isalso bounded by (18) and Theorem 4.1 for X ∈ P ( E ) k . We also haveΨ F ( X ) = B , ⊗ I + k X i =1 B i, ⊗ ( X i − I ) − " B , ⊗ I + k X i =1 B i, ⊗ ( X i − I ) × " B , ⊗ I + k X i =1 B i, ⊗ ( X i − I ) − × " B , ⊗ I + k X i =1 B i, ⊗ ( X i − I ) where B i,xy := M dim( E ) < ∞ M ( A,v ) ∈ Ω c ,c B i,xy ( A, v )for 0 ≤ i ≤ k and x, y ∈ { , } . Lemma 4.5. Let F : P k P be an operator monotone function and let dim( E ) < ∞ . Let A j ∈ P c ,c ( E ) k and v j ∈ S ( E ) for j ∈ J for some finiteindex set J . Then there exists a w ∈ S ( H ) such that (25) F ( A j ) v j = ( w ∗ ⊗ I )Ψ F ( A j )( w ⊗ I ) v j for all j ∈ J . ¨OWNER’S THEOREM IN SEVERAL VARIABLES 27 Proof. Let A := L j ∈J A j and v := L j ∈J √ |J | v j . Then by Theorem 4.4and the definition of Ψ F , we have F ( A ) v = v ∗ B , ( A, v ) v ⊗ Iv + k X i =1 v ∗ B i, ( A, v ) v ⊗ ( A i − I ) v − ( ( v ∗ ⊗ I ) " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) × " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) − × " B , ( A, v ) ⊗ I + k X i =1 B i, ( A, v ) ⊗ ( A i − I ) ( v ⊗ I ) ) v = ( w ∗ ⊗ I )Ψ F ( A )( w ⊗ I ) v, where w := ( · · · ⊕ ⊕ v ⊕ ⊕ · · · ) ∈ S ( H ), and the nonzero v is at theappropriate coordinate such that the above holds, according to the definitionof Ψ F . Since both ( A, v ) F ( A ) v and ( A, v ) ( w ∗ ⊗ I )Ψ F ( A )( w ⊗ I ) v are natural maps, we have (25). (cid:3) Let B +1 ( H ) ∗ denote the state space of B ( H ) and B +1 ( H ) ∗ its normal part.Note that positive linear functionals on unital C ∗ -algebras attain their normat the unit, hence B +1 ( H ) ∗ is convex, weak- ∗ compact by Banach-Alaoglu. Theorem 4.6. Let F : P k P be an operator monotone function. Thenthere exists a w ∈ B +1 ( H ) ∗ such that for all dim( E ) < ∞ and X ∈ P c ,c ( E ) k we have F ( X ) =( w ⊗ I )(Ψ F ( X ))= w ( B , ) ⊗ I + k X i =1 w ( B i, ) ⊗ ( X i − I ) − ( w ⊗ I ) (" B , ⊗ I + k X i =1 B i, ⊗ ( X i − I ) × " B , ⊗ I + k X i =1 B i, ⊗ ( X i − I ) − × " B , ⊗ I + k X i =1 B i, ⊗ ( X i − I ) . (26) Proof. For A ∈ P c ,c ( E ) k and v ∈ E by Lemma 4.5 the set L ( A,v ) := { w ∈ B +1 ( H ) ∗ : ( w ⊗ I )(Ψ F ( A )) v = F ( A ) v } is nonempty, moreover it is easy to check that it is a closed subset of B +1 ( H ) ∗ in the weak- ∗ topology of B ( H ) ∗ hence compact. Indeed, the latter is theconsequence of the identification B ( H ⊗ E ) = B ( H dim( E ) ) ≃ M dim( E ) ( B ( H )),where M dim( E ) ( B ( H )) denotes the dim( E )-by-dim( E ) matrices with B ( H )valued entries.Let L denote the collection { L ( A,v ) : A ∈ P c ,c ( E ) k , v ∈ E } of subsets of B +1 ( H ) ∗ . Any finite sub-collection from L has the form { L ( A,v ) : ( A, v ) ∈ Ω nc ,c } for some integer n , so by Lemma 4.5 has nonempty intersection. Thismeans that L has the finite intersection property. So the weak- ∗ compactnessof B +1 ( H ) ∗ implies that there is a w ∈ B +1 ( H ) ∗ which is in every L ( A,v ) ,proving the assertion. (cid:3) Analytic properties of resolvents Definition 5.1. The imaginary part of a bounded linear operator A isdefined as ℑ A := A − A ∗ i , and its real part is ℜ A := A + A ∗ . Proposition 5.1. Let A := (cid:20) A A A A (cid:21) be a block operator matrix with ℑ A ≥ . Then if its Schur complement S ( A ) = A − A A − A exists, it also satisfies ℑS ( A ) ≥ .Similarly if ℜ A ≥ and if its Schur complement exists, it also satisfies ℜS ( A ) ≥ .Proof. By assumption we have0 ≤ ℑ (cid:26) ( x ⊕ x ) ∗ (cid:20) A A A A (cid:21) ( x ⊕ x ) (cid:27) for any vector ( x ⊕ x ). Hence choosing x = − A − A x we get that0 ≤ ℑ (cid:26) ( x ⊕ x ) ∗ (cid:20) A A A A (cid:21) ( x ⊕ x ) (cid:27) = ℑ (cid:26) ( x ⊕ x ) ∗ (cid:20) A − A A − A (cid:21)(cid:27) = ℑ{ x ∗ S ( A ) x } proving the first part of the assertion. The second part covering the realparts is proved similarly. (cid:3) The following result for the Schur complement is also well known and itsproof can be found for example in [7]. ¨OWNER’S THEOREM IN SEVERAL VARIABLES 29 Proposition 5.2. Let A := (cid:20) A A A A (cid:21) and B := (cid:20) B B B B (cid:21) be conformally partitioned, positive semi-definite block operator matrices.Then if A ≤ B then also S ( A ) ≤ S ( B ) , i.e. the Schur complement isoperator monotone.Moreover S ( · ) is also operator concave, i.e. (1 − λ ) S ( A ) + λ S ( B ) ≤S ((1 − λ ) A + λB ) for all λ ∈ [0 , . Definition 5.2 (Sectorial operator) . For A ∈ B ( E ) let W ( A ) := { x ∗ Ax : x ∈ E, k x k = 1 } denote the numerical range of A . We say that A is sectorialif W ( A ) ⊆ S α , where S α := { z ∈ C : ℜ ( z ) > , |ℑ ( z ) | ≤ Re ( z ) tan α } forsome α ∈ [0 , π/ X ∈ B ( N ) where dim( N ) < ∞ let { σ j ( X ) } ≤ j ≤ dim( N ) denotethe ordered decreasing sequence of its singular values. Proposition 5.3 (Theorem 1.1 [17]) . Let A ∈ B ( N ) with dim( N ) < ∞ and W ( A ) ⊆ S α be partitioned as A := (cid:20) A A A A (cid:21) . Let S ( A ) = A − A A − A . Then σ j ( S ( A )) ≤ sec ( α ) σ j ( A ) . In particular under the assumptions above including also dim( N ) = ∞ andthat A has a bounded inverse, we have (27) kS ( A ) k ≤ sec ( α ) k A k . Proof. The proof of this under the assumption dim( N ) < ∞ can be found in[17]. Invertibility follows from W ( A ) ⊆ S α and the result in [17] covers theinequality for the singular values and from that (27) follows for the finitedimensional case, since k A k ≤ k A k .If dim( N ) = ∞ , we assume that the block operator matrix A is partitionedaccording to the orthogonal decomposition N = S ⊕ S ⊥ , where S is a closedsubspace of N . We can approximate A in the strong operator topologyby a net of finite rank operators P γ AP γ , where P γ is the directed set ofprojections with ranges running over all finite dimensional subspaces of N partially ordered under subspace inclusion. Indeed, A γ := P γ AP γ convergesto A in the strong operator topology, since k P γ AP γ x − Ax k ≤ k P γ AP γ x − P γ Ax k + k P γ Ax − Ax k≤ k P γ A kk P γ x − x k + k P γ Ax − Ax k≤ k A kk P γ x − x k + k P γ Ax − Ax k and k P γ y − y k → y ∈ N by construction. Notice that A γ = P γ AP γ is also sectorial on the closed subspace P γ N ≤ N and existence of the Schurcomplement S ( A γ ) is justified by Theorem 1 in [5].We claim that ( A γ ) − → A − in the strong operator topology. To seethis let P S denote the orthogonal projection such that A = P S AP S . Then( A γ ) = P S P γ AP γ P S ⊆ S α on P γ P S N ≤ N . Then for any z ∈ P γ P S N with k z k = 1 we have(28) | z ∗ ( A γ ) z | ≤ k z kk ( A γ ) z k = k ( A γ ) z k . Since W (( A γ ) ) is a closed and compact subset of the open sector S α , wehave that | W (( A γ ) ) | has a greatest lower bound c > | W (( A γ ) ) | . Hence from (28) we have that ( A γ ) is lower bounded,i.e. k ( A γ ) z k ≥ c k z k for any z ∈ P γ P S N , hence ( A γ ) is invertibleon P γ P S N ≤ N and we have ( A γ ) − = (( A γ ) ∗ ( A γ ) ) − ( A γ ) ∗ . Simi-larly since A is invertible on P S N ≤ N it is lower bounded and A − =( A ∗ A ) − A ∗ . We also have that k ( A γ ) ∗ ( A γ ) k = k P S P γ A ∗ P γ P S P S P γ AP γ P S k≤ k A ∗ P γ P S P S P γ A k≤ k A ∗ A k where to obtain the last inequality we used that I ≥ P γ P S P S P γ . Similarly k A ∗ A k ≤ k A ∗ A k , hence it follows that12 I P S N ≥ k A ∗ A k A ∗ A =: X I P γ P S N ≥ k A ∗ A k ( A γ ) ∗ ( A γ ) =: X γ , moreover we have that X and X γ are lower bounded. Hence we have forsome 1 > t, t γ > k I P S N − X k < t k I P γ P S N − X γ k < t γ , so it follows that X − = [ I P S N − ( I P S N − X )] − = ∞ X l =0 ( I P S N − X ) l ,X − γ = (cid:2) I P γ P S N − (cid:0) I P γ P S N − X γ (cid:1)(cid:3) − = ∞ X l =0 (cid:0) I P γ P S N − X γ (cid:1) l (29) ¨OWNER’S THEOREM IN SEVERAL VARIABLES 31 where I P S N and I P γ P S N are the identity operators on the respective Hilbertsubspaces P S N and P γ P S N . Since operator multiplication is jointly strongoperator continuous on bounded sets, we have X γ → X and I P γ P S N → I P S N ,hence(30) (cid:0) I P γ P S N − X γ (cid:1) l → ( I P S N − X ) l in the strong operator topology for all l ≥ 0, since also P γ → I in the strongoperator topology. The sums in (29) are uniformly convergent in the normtopology, hence also in the strong operator topology, so it follows by (30)that X − γ → X − and then (( A γ ) ∗ ( A γ ) ) − → ( A ∗ A ) − in the strongoperator topology. We have that ( A γ ) ∗ = P S P γ A ∗ P γ P S → P S A ∗ P S = A ∗ in the strong operator topology, then by the joint strong operator continuityof operator multiplication on bounded sets we get that ( A γ ) − → A − inthe strong operator topology as claimed.Using the claim that ( A γ ) − → A − in the strong operator topology andthe joint strong operator continuity of operator multiplication on boundedsets we obtain S ( A γ ) → S ( A )where S ( A ) = A − A A − A and S ( A γ ) = ( A γ ) − ( A γ ) ( A γ ) − ( A γ ) .By (27) kS ( A γ ) k ≤ sec ( α ) k A γ k and since k A γ k ≤ k A k we have kS ( A γ ) k ≤ sec ( α ) k A k . Since S ( A γ ) z → S ( A ) z for any vector z we have kS ( A γ ) z k → kS ( A ) z k ,hence kS ( A ) z k ≤ sec ( α ) k A k for any k z k = 1 and we obtain (27) in the infinite dimensional case aswell. (cid:3) Lemma 5.4. Let L B ( X ) = P ki =1 B i ⊗ X i be a linear matrix pencil over theHilbert space K ⊗ E such that B i ≥ . If all X i are sectorial, then L B ( X ) is also sectorial on N ⊥ ( L B (1)) ⊗ E , where N ⊥ ( L B (1)) denotes the closureof the complement of the kernel of L B (1) = P ki =1 B i ⊗ P ki =1 B i .Suppose additionally to the above that P ki =1 B i is lower bounded on N ⊥ ( L B (1)) .Then if all ℜ X i is lower bounded, then ℜ L B ( X ) is also lower bounded on N ⊥ ( L B (1)) ⊗ E .Proof. We begin with a simple observation. If A ∈ B ( N ) is a sectorialoperator on a Hilbert space N , then for any X ∈ B ( N ) the operator X ∗ AX is also sectorial. Now for γ = P j ∈I e ∗ j ⊗ γ j ∈ N ⊥ ( L B (1)) ⊗ E we have γ ∗ L B ( X ) γ = k X i =1 tr { B i Γ ∗ X i Γ } = k X i =1 tr { B / i Γ ∗ X i Γ B / i } = k X i =1 X j ∈I e ∗ j B / i Γ ∗ X i Γ B / i e j where Γ = P j ∈I γ j e ∗ j and convergence is in the ultraweak operator topology,tr Γ ∗ Γ = k γ k and { e j } j ∈I denotes an orthonormal basis of N ⊥ ( L B (1)). Fromthe above it follows that W ( L B ( X )) ⊆ S α if X i ⊆ S α for all 1 ≤ i ≤ k anda fixed α ∈ [0 , π/ X i is sectorial,we have that ℜ X i ≥ 0. Moreover each ℜ X i is lower bounded, hence thereexists a real number ǫ > ≤ i ≤ k we have ℜ X i ≥ ǫI .Then we have that(31) ℜ L B ( X ) = k X i =1 B i ⊗ ℜ X i ≥ k X i =1 B i ⊗ ǫI. Since P ki =1 B i is lower bounded on N ⊥ ( L B (1)), it follows from (31) that ℜ L B ( X ) is lower bounded on N ⊥ ( L B (1)) ⊗ E as well. (cid:3) Definition 5.3 (Operator poly-halfspaces) . For a Hilbert space E the upperoperator poly-halfspace is defined asΠ k := { X ∈ B ( E ) k : ℑ X i > , ≤ i ≤ k } , while the right operator poly-halfspace asΣ k := { X ∈ B ( E ) k : ℜ X i > , ≤ i ≤ k } . We also use the notation Π := Π and Σ := Σ . Proposition 5.5. Let L B ( X ) = B ⊗ I + P ki =1 B i ⊗ X i be a linear matrixpencil over the Hilbert space K ⊗ E such that B i ≥ . Assume that both B and P ki =1 B i are lower bounded on N ⊥ ( B ) and N ⊥ ( P ki =1 B i ) respectively.Let S be a Hilbert subspace of K and let P S ∈ B ( K ) denote the orthogonalprojection onto S . Then for each X ∈ Σ k or X ∈ Π k the Schur complement S ( L B ( X )) = P L B ( X ) P − P L B ( X ) P ⊥ h P ⊥ L B ( X ) P ⊥ i − P ⊥ L B ( X ) P exists, where P = P S ⊗ I , P ⊥ = ( I − P S ) ⊗ I , moreover there exists an α ∈ [0 , π/ depending on X but independent of B i such that (32) kS ( L B ( X )) k ≤ sec α k L B ( X ) k . ¨OWNER’S THEOREM IN SEVERAL VARIABLES 33 Proof. First assume that X ∈ Σ k . Then there exists an ǫ > ℜ X i ≥ ǫI . Then ℜ L B ( X ) = B ⊗ I + k X i =1 B i ⊗ ℜ X i ≥ B ⊗ I + k X i =1 B i ⊗ ǫI = B + ǫ k X i =1 B i ! ⊗ I, hence ℜ L B ( X ) is lower bounded on N ⊥ k X i =0 B i ! ⊗ E = N ⊥ ( B ) ∪ N ⊥ k X i =1 B i !! ⊗ E, thus ℜ L B ( X ) is invertible on N ⊥ ( P ki =0 B i ) ⊗ E . Since ℑ L B ( X ) = P ki =1 B i ⊗ℑ X i it follows that N ⊥ ( ℑ L B ( X )) ≤ N ⊥ ( ℜ L B ( X )). Thus L B ( X ) = ℜ L B ( X ) + i ℑ L B ( X )= ( ℜ L B ( X )) / [ I + i ( ℜ L B ( X )) − / ℑ L B ( X )( ℜ L B ( X )) − / ]( ℜ L B ( X )) / is lower bounded on N ⊥ ( P ki =0 B i ) ⊗ E , since for any A ∈ B ( N ⊥ ( ℑ L B ( X )) ⊗ E ) we have ( I + iA ) ∗ ( I + iA ) = I + A ∗ A which is lower bounded. It followsthat P ⊥ L B ( X ) P ⊥ is lower bounded as well on P ⊥ (cid:16) N ⊥ ( P ki =0 B i ) ⊗ E (cid:17) ,thus invertible and this implies that S ( L B ( X )) exists and is bounded. Since X ∈ Σ k and ℜ X i ≥ ǫI , there exists an α ∈ [0 , π/ 2) such that W ( X i ) ⊆ S α for all 1 ≤ i ≤ k . Notice that W ( I ) ⊆ S α as well. Then by Lemma 5.4 wehave that W ( L B ( X )) ⊆ S α and Proposition 5.3 implies (32).Now assume that X ∈ Π k . Then there exists an ǫ > ℑ X i ≥ ǫI .Then there exists an c > ℑ L B ( X ) = k X i =1 B i ⊗ ℑ X i ≥ k X i =1 B i ⊗ ǫI N ⊥ ( P ki =1 B i ) ≥ c I N ⊥ ( P ki =1 B i ) ⊗ I, hence ℑ L B ( X ) is lower bounded on N ⊥ (cid:16)P ki =1 B i (cid:17) ⊗ E . There also exists a c ∈ R such that ℜ L B ( X ) − B ⊗ I = P ki =1 B i ⊗ ℜ X i ≥ c I N ⊥ ( P ki =1 B i ) ⊗ I .Then there exists a θ ∈ ( − π/ , 0] such that(33) ℜ e iθ ℜ L B ( X ) − ℑ e iθ ℑ L B ( X ) > , equivalently lower bounded, on ( N ⊥ ( B ) ∪ N ⊥ ( P ki =1 B i )) ⊗ E . Indeed, since ℜ L B ( X ) ≥ B ⊗ I + c I N ⊥ ( P ki =1 B i ) ⊗ I and ℑ L B ( X ) ≥ c I N ⊥ ( P ki =1 B i ) ⊗ I and B ≥ N ⊥ ( B ), it suffices to have c ℜ e iθ − c ℑ e iθ > . Then e iθ I and each e iθ X i for all 1 ≤ i ≤ k is sectorial for some α ∈ [0 , π/ W ( e iθ I ) , W ( e iθ X i ) ⊆ S α . Thus Lemma 5.4 implies that W ( e iθ L B ( X )) ⊆ S α and (33) implies that ℜ ( e iθ L B ( X )) is lower bounded. Now a similar argu-ment as in the previous case implies that P ⊥ e iθ L B ( X ) P ⊥ is lower boundedon P ⊥ (cid:16) ( N ⊥ ( B ) ∪ N ⊥ ( P ki =1 B i )) ⊗ E (cid:17) as well, thus invertible and this im-plies that S ( e iθ L B ( X )) exists and is bounded, which implies the existenceand boundedness of S ( L B ( X )) as well, since S ( e iθ L B ( X )) = e iθ S ( L B ( X )).Then Proposition 5.3 implies (32) with e iθ L B ( X ) in place of L B ( X ). Then(32) follows as well.The above argumentation excludes the case when S and N ⊥ ( B ) ∪N ⊥ ( P ki =1 B i )have trivial intersection. However in this case there is nothing to prove sincethen we have S ( L B ( X )) = L B ( X ) and (32) is satisfied with α = 0. (cid:3) Now we may perform the free analytic continuation of our free function F ( X ) = ( w ⊗ I )(Ψ F ( X )) given in Theorem 4.6. Proposition 5.6. Let F : P k P be an operator monotone function andlet c > c > be real numbers. Let w ∈ B +1 ( H ) be a state in Theorem 4.6such that for all dim( E ) < ∞ and X ∈ P c ,c ( E ) k we have (34) F ( X ) = ( w ⊗ I )(Ψ F ( X )) . Then F has a free analytic continuation ˜ F to the whole of Σ k and Π k forany Hilbert space E , such that F maps Π k to Π , moreover ˜ F is operatormonotone and operator concave on P ( E ) k and maps P ( E ) k to P ( E ) .Proof. First we take care of the analytic continuation to Σ k . Notice that ac-cording to Theorem 4.4 and Proposition 3.8 each direct summand of Ψ F ( X )is the Schur complement of a linear pencil of the form L F,A,v ( X ) := B ( F, A, v ) ⊗ I + k X i =1 B ( F, A, v ) i ⊗ ( X i − I ) . For each L F,A,v we have that B ( F, A, v ) i ∈ B + ( N F,A,v ) ∗ for some finite di-mensional Hilbert subspace N F,A,v ≤ H , P ki =1 B ( F, A, v ) i ≤ B ( F, A, v ) and tr { B ( F, A, v ) } ≤ F ( c ,...,c )min(1 ,c ) by Proposition 3.8. For a fixed X ∈ Σ k or X ∈ Π k it is easy to find a uniform upper bound on the norms ofall L F,A,v ( X ) since we have the bound tr { B ( F, A, v ) } ≤ F ( c ,...,c )min(1 ,c ) and B ( F, A, v ) i ∈ B + ( H ) ∗ , P ki =1 B ( F, A, v ) i ≤ B ( F, A, v ) . Then for all A and v separately, Proposition 5.5 applies for L F,A,v ( X ) with subspace S spannedby v . Then according to Definition 4.2 Ψ F ( X ) is the direct sum of all theSchur complements S ( L F,A,v ( X )), each with norm bounded from above bya uniform constant. HenceΨ F ( X ) ∗ Ψ F ( X ) ≤ K sec αI ⊗ I for some large enough real constant K > X . Notice that( w ⊗ I ) : B ( H ) ⊗ B ( E ) C ⊗ B ( E ) ∼ = B ( E ) is a completely positive unital ¨OWNER’S THEOREM IN SEVERAL VARIABLES 35 linear map. Hence by the Schwarz inequality for 2-positive unital linearmaps, see Proposition 3.3 [51], we have( w ⊗ I )(Ψ F ( X )) ∗ ( w ⊗ I )(Ψ F ( X )) ≤ ( w ⊗ I )(Ψ F ( X ) ∗ Ψ F ( X )) ≤ K sec αI, hence (34) defines a free holomorphic/analytic function on Σ k and Π k . Thenby Proposition 5.1 ˜ F maps Π k to Π and by Proposition 5.2 ˜ F is operatormonotone and operator concave on P ( E ) k and clearly maps P ( E ) k to P ( E ). (cid:3) Theorem 5.7. Let E be a Hilbert space and let F : P ( E ) k P ( E ) be a freefunction. Then the following are equivalent: (a) F is operator monotone; (b) F is operator concave; (c) There exists a Hilbert space K , a closed subspace K ≤ K and thecorresponding orthogonal projection P K with range K , B i ∈ ˆ P ( K ) , ≤ i ≤ k with B ≥ P ki =1 B i and a state w ∈ B +1 ( K ) ∗ such that forall X ∈ P ( E ) k we have F ( X ) = w ( B , ) ⊗ I + k X i =1 w ( B i, ) ⊗ ( X i − I ) − ( w ⊗ I ) (" B , ⊗ I + k X i =1 B i, ⊗ ( X i − I ) B , ⊗ I + k X i =1 B i, ⊗ ( X i − I ) − " B , ⊗ I + k X i =1 B i, ⊗ ( X i − I ) (35) where B i, ( A, v ) := P K B i P K ,B i, ( A, v ) := P K B i ( I − P K ) ,B i, ( A, v ) :=( I − P K ) B i P K ,B i, ( A, v ) :=( I − P K ) B i ( I − P K );(36)(d) F has a free analytic continuation to Π k , mapping Π k to Π .Proof. The equivalence between (a) and (b) is given by Theorem 2.5 andCorollary 2.4.That (c) implies the others is given by Proposition 5.1, Proposition 5.2and by Proposition 5.6.The proof that (d) implies (a) can be found in [50] as Lemma 4.8, weprovide here the proof for completeness. First of all, similarly to the one variable case, a differentiable free function F is operator monotone if andonly if its Fr´echet-derivative DF ( X )( H ) ≥ X in the domain of F and any tuple H ≥ 0. Indeed if F is operator monotone then DF ( X )( H ) = lim t → F ( X + tH ) − F ( X ) t and by monotonicity F ( X + tH ) ≥ F ( X ) for H ≥ t ≥ 0, hence DF ( X )( H ) ≥ 0. Conversely F ( X ) − F ( Y ) = Z DF ((1 − t ) Y + tX )( X − Y ) dt, hence if X ≥ Y and DF ( Z )( H ) ≥ H ≥ Z in the domain of F ,then the integral on the right is positive semidefinite, hence F ( X ) ≥ F ( Y ) aswell, establishing that F is operator monotone. To show that (d) implies (a)assume that (d) holds, but F is not operator monotone. Then by the above DF ( X )( H ) is not positive semidefinite for some X ∈ P ( E ) k and H ≥ H ∈ S ( E ) k . Since F ( X + itH ) = F ( X ) + itDF ( X )( H ) + O ( t ) , ℑ F ( X + itH ) = tDF ( X )( H ) + O ( t ) , for this X and H it follows that ℑ F ( X + itH ) is not positive semidefinitefor small enough t ≥ 0, a contradiction.Lastly we show that (a) implies (c). For any fixed c > c > 0, by Theo-rem 4.6 (c) holds for all dim( E ) < ∞ and X ∈ P c ,c ( E ) k . We use Proposi-tion 5.6 to free analytically continue F to Σ k . We can follow this procedurefor any c > c > 0. Now we have identity and uniqueness theorems fornon-commutative power series expansions for free analytic functions, seeTheorem 7.2, 7.8 and 7.9 in [58]. The non-commutative power series ex-pansion around an arbitrary point A ∈ Σ k is called a TT series in [58]and the coefficients of TT series are uniquely determined by the directionalderivatives of the free function at A , and the series uniformly converges onsome open ball around A . Therefore for each c > c > k by expanding the Schur complement intoa non-commutative power series converging uniformly on open balls aroundarbitrary points in A ∈ Σ k representing the functions in (35). Hence forany c > c > k , and by similar arguments on Π k as well. Thus we can pickany one them of the form as in (c) for some fixed c > c > E ) < ∞ . Now assume that E is separable. Then by Lemma 3.4 F isstrong operator continuous, hence for any bounded from above increasingnet of k -tuple of operators { A i } i ∈I with A i ∈ P ( E ) k we have(37) sup i ∈I F ( A i ) = F (cid:18) sup i ∈I A i (cid:19) . ¨OWNER’S THEOREM IN SEVERAL VARIABLES 37 If E is non-separable, then by Assumption 1 for any bounded from aboveincreasing net of k -tuple of operators { A i } i ∈I with A i ∈ P ( E ) k we havesup i ∈I F ( A i ) ≥ F (cid:18) sup i ∈I A i (cid:19) , which implies (37), since F ( A i ) ≤ F (sup i ∈I A i ) by operator monotonicity of F alone. Now using this strong continuity we can approximate any tuple X ∈ P ( E ) k in the strong product topology by a bounded from above increasingnet of tuples of finite rank operators X γ ∈ P ( E ) k , such that X γ → X strongly. For each F ( X γ ) (c) holds if we restrict each X γ to the orthogonalcomplement of its kernel. Notice that a free function defined by the formula(35) is strong operator continuous on norm bounded sets since it can beexpanded into a uniformly norm-convergent non-commutative power series.The other way to see this is by Lemma 3.4 and the norm continuity ofthe formula (35). Then by the strong operator continuity of F , we have F ( X γ ) → F ( X ) in the strong operator topology, hence (c) holds for any E and X ∈ P ( E ) k . (cid:3) Now given an arbitrary operator monotone NC function F : S a ,b ( E ) ×· · · × S a k ,b k S where S a,b := { X ∈ S ( E ) : aI ≤ X i ≤ bI } for real numbers b > a , we can apply Theorem 5.7 to free analytically continue the functionusing the following transformations. M¨obius transformations(38) g ( x ) = ax + bcx + d are operator monotone for x ∈ R , x = − d/c provided ad − bc > g which maps ( a, b ) to (0 , ∞ ) bijectively, with g (and also its inverse) being operator monotone. Hence we may composewith such operator monotone transformations for each variable of F , toobtain a new operator monotone NC function ˆ F : P k S from F . If ˆ F isbounded from below then after adding some cI for real c ≥ F + cI > k to Π. If ˆ F is not bounded from below on P k , then F ( X ) := ˆ F ( X + ǫI ) is bounded from below on P k by operator monotonicityfor any real ǫ > 0, hence F satisfies the previous case. In this way we obtainthe free analytic continuation of the original F mapping Π k to Π. Acknowledgment An initial version of this paper appeared online in May 2014, where theauthor claimed more or less similar results stated in this paper. Howeverthe proofs contained a major flaw. The flaw was found later by the author,however theoretical counterexamples were communicated to the author ear-lier by James E. Pascoe and certain numerical counterexamples by Takeaki Yamazaki. The author would like to express his gratitude to James E. 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