Analytic continuation of concrete realizations and the McCarthy Champagne conjecture
aa r X i v : . [ m a t h . F A ] S e p ANALYTIC CONTINUATION OF CONCRETEREALIZATIONS AND THE MCCARTHY CHAMPAGNECONJECTURE
KELLY BICKEL, J. E. PASCOE, AND RYAN TULLY-DOYLE
Abstract.
In this paper, we give formulas that allow one to movebetween transfer function type realizations of multi-variate Schur,Herglotz and Pick functions, without adding additional singular-ities except perhaps poles coming from the conformal transfor-mation itself. In the two-variable commutative case, we use acanonical de Branges-Rovnyak model theory to obtain concreterealizations that analytically continue through the boundary forinner functions which are rational in one of the variables (so-called quasi-rational functions ). We then establish a positive solutionto McCarthy’s Champagne conjecture for local to global matrixmonotonicity in the settings of both two-variable quasi-rationalfunctions and d -variable perspective functions. Contents
1. Introduction 21.1. Overview 21.2. Background 31.3. Summary of results 72. Two-variable realization formulae 83. Boundary behavior of quasi-rational functions 154. Some algebraic identities 244.1. The block 2 by 2 matrix inverse formula 25
Date : September 30, 2020.Bickel was supported in part by NSF-DMS Analysis grant
Introduction
Overview.
Colloquially, realizations are ways of representing struc-tured classes of functions using operators on a Hilbert space; thesebridges between rich operator-theoretic results and concrete functiontheory have led to a myriad of important breakthroughs. Classicrealizations-type formulae include the Nevanlinna representations for
Pick functions (holomorphic functions mapping the upper half planeΠ to Π) and the transfer function realizations for
Schur functions (holomorphic functions mapping the unit disk D to D ).In [2], J. Agler extended such one-variable formulae from systems en-gineering into functional analysis in several variables; this heralded ina period of rapid development for function theory on the bidisk D andpolydisk D d , including extensions of Pick interpolation, the infinites-imal Schwarz lemma, L¨owner’s theorem, and the Julia-Carath´eodorytheorem [1, 34, 4, 5, 47]. Realization theory has also been extended tononcommutative functional analysis, an area that has seen an explo-sion of activity in the last decade. Specifically, J. Williams developeda realization theory in the free probability setting in [59]. In the freeanalysis setting, realizations for free Pick functions were developed in[51, 48], which is part of a large body of recent and ongoing work invarious noncommutative contexts [16, 8, 12, 35, 53, 54, 43, 42, 10, 11]. ONCRETE REALIZATIONS AND THE MCC 3
As in the commutative case, these realizations can be used to gen-eralize classical theorems of complex analysis to functions of severalnoncommuting variables.This paper investigates three foundational questions that one canask about general realizations:Q1: When do the regularity properties of a realization exactly mimicthose of the represented function?Q2: How does one move between realization formulae without sac-rificing fine behavior?Q3: Are there settings where realizations possess identifiable con-crete formulae?In this paper, we use functional analysis on the bidisk to answer (Q1)and (Q3) for classes of two-variable Schur functions. We also develop amore general algebraic approach to (Q2), which yields a chain of oper-ator expressions that relates Schur, Herglotz, and Pick-type structuresand is applicable to the noncommutative setting. We then provideapplications in the context of several variable functional analysis.1.2.
Background.
To motivate this investigation, consider the one-variable setting, and recall that Pick functions f : Π → Π can bewritten uniquely in the following form, called a
Nevanlinna repre-sentation , f ( z ) = a + bz + Z R tzt − z d µ ( t )for some a ∈ R , b ∈ R ≥ , and µ a positive finite Borel measure on R [44, 39]. The complement of the support of µ is exactly the set where f analytically continues to be real valued, and thus through the realline via the Schwarz reflection principle. A similar fact holds for theearlier classical Herglotz integral representation for functions from thedisk to the right half plane [31, 56]. Nevanlinna and Herglotz functionshave a number of applications, for example to the study of finite rankperturbations of self-adjoint operators; see the survey papers [40, 41],book [33] and references within. KELLY BICKEL, J. E. PASCOE, AND RYAN TULLY-DOYLE
Similarly, Schur functions φ : D → D possess a transfer functionrealization (or TFR); i.e. they can be written in form φ ( z ) = A + B (1 − zD ) − zC for z ∈ D , where U = " A BC D : " C M → " C M is a contraction on a Hilbert space C ⊕ M , see [30]. The operator U can be chosen to be isometric, coisometric, or unitary; in each case,the choice is unique up to certain minimality assumptions and unitaryequivalence, and there are concrete function theory interpretations forthe canonical Hilbert spaces M and the operators A, B, C, D , see [22,14, 7]. Under minimality assumptions, the set of τ ∈ T where 1 − τ D isinvertible is exactly the set where φ analytically continues with modulus1 and can therefore be analytically continued via the reflection principleon the disk.The pioneering work of Agler in [2] (part of which was independentlyestablished by Kummert in [38]) implies that each Schur function φ : D → D has a two-variable TFR and hence can be written as φ ( z ) = A + B (1 − E z D ) − E z C for z ∈ D , where U = " A BC D : " C M → " C M is a contraction on a Hilbert space C ⊕ M and can be chosen to beunitary, isometric, or coisometric. Here M decomposes as M ⊕ M and E z = z P + z P where each P j is the projection onto M j .While Agler’s initial proof was nonconstructive, influential work byBall, Sadosky, and Vinnikov in [17] used minimal scattering systems(for example, the so-called de Branges-Rovnyak model associated to Here and throughout the paper, “1” denotes the identity operator on an ap-propriate Hilbert space that should be clear from the context. The notation “ I ” isreserved for an interval or open set. ONCRETE REALIZATIONS AND THE MCC 5 φ ) and concrete Hilbert space geometry to produce and analyze morespecific TFRs. They continued this seminal work with Kaliuzhnyi-Verbovetskyi in [15], which includes an exhaustive analysis of TFRsand connections between the geometric scattering structure and asso-ciated formal reproducing kernel Hilbert spaces. Ball and Bolotnikovconducted additional insightful work on canonical TFRs in [13, 14].Many of these references also include results for the more general Schur-Agler class on D d , and we refer the reader to [29] for interesting relatedresults concerning general Schur functions on D d . If a Schur function φ is inner , i.e. iflim r ր | φ ( rτ ) | = 1 for a.e. τ ∈ T , then the Hilbert space geometry from [17] simplifies dramatically. In-deed, in [20, 21], the first author and G. Knese constructed particu-larly simple coisometric TFRs for two-variable inner functions using theBall-Sadosky-Vinnikov machinery from [17]. This methodology yieldedexplicit formulae for the (reproducing kernel) Hilbert space M , regular-ity properties of the functions in M , and information about A, B, C, D .If φ is both rational and inner, then its TFRs come directly from sumsof squares decompositions of related stable polynomials, i.e. polyno-mials that do not vanish on D , see [20, 38, 28, 35, 60]. In this case, φ possesses a minimal TFR in the sense that if the degree of φ in z j is m j , then M can be chosen so dim M = m + m . The proof ofthis minimality result is embedded in Kummert’s work [38], and anextension with particularly clear exposition can be found in [36].The above minimality result was a key tool in [5]. In this ground-breaking paper, Agler, McCarthy, and Young characterized multivari-ate monotone matrix functions via two types of monotonicity, a global condition and a local condition. Specifically, a real-valued func-tion f is globally matrix monotone on an open set E ⊆ R d if forany positive integer n and any pair of d -tuples of commuting n × n self-adjoint matrices A = ( A , . . . , A d ), B = ( B , . . . , B d ) with each KELLY BICKEL, J. E. PASCOE, AND RYAN TULLY-DOYLE A j ≤ B j and joint spectrum in E , one has f ( A ) ≤ f ( B ). Meanwhile, f is called locally matrix monotone on E if the previous-describedrelation holds on positively-oriented paths in the variety of commutingself-adjoint matrices. As the exact notation of local matrix monotonic-ity is cumbersome and not required in the current discussion, we referthe reader to [5, 47] for details.The work in [5] with later minor refinements in [47] yields the fol-lowing characterization of local matrix monotonicity: Theorem 1.1. [5] . Let E be an open set in R d . A function f : E → R is locally matrix monotone on E if and only if f analytically continuesto Π d as a map f : E ∪ Π d → Π in the Pick-Agler class. When d = 1 or d = 2, the Pick-Agler class is exactly the Pick class,i.e. the set of analytic functions mapping Π d to Π. More generally, thePick-Agler class is the set of Pick functions that satisfy von Neumann’sinequality after being converted to Schur functions via conformal map-pings. The two-variable von Neumann inequality is known as Andˆo’sinequality [9] and fails in more than two variables [45, 58]. For d > Theorem 1.2. [5] . If E ⊆ R is a rectangle and f : E → R is rational,then f is globally matrix monotone on E if and only if f analyticallycontinues to Π as a map f : E ∪ Π → Π in the Pick class. The question of whether real-valued restrictions of Pick-Agler func-tions to convex sets in R d are always global matrix monotone functionshas colloquially become known as the McCarthy Champagne conjec-ture: ONCRETE REALIZATIONS AND THE MCC 7 (MCC): Every d -variable Pick-Agler function thatanalytically continues across an open convex set E ⊆ R d isglobally matrix monotone when restricted to E . As the discussed further below, we establish the MCC in two impor-tant cases, giving compelling evidence for the overall validity of theconjecture.1.3.
Summary of results.
The bulk of this paper addresses the re-alization questions (Q1)-(Q3). In Section 2, we let φ be a two-variableinner function, review the particularly simple TFRs from [20, 21], andfurther develop their properties. For example, in Theorem 2.1, we ex-tend the analysis from [21] to answer (Q3) and provide explicit formulaefor each of A, B, C, D .This allows us to address (Q1) for quasi-rational functions in Section3. Here, we say that a two-variable Schur function φ is quasi-rational with respect to an open I ⊆ T if φ is inner and extends continuouslyto T × I with | φ ( τ ) | = 1 for τ ∈ T × I . The analysis from both Section2 and [20, 21] allows us to establish this key regularity property: Theorem. φ is quasi-rational with respect to I and D is fromTheorem 2.1, then 1 − E τ D is invertible for all τ ∈ T × I .It is worth noting that this question of when operators of the form(1 − E τ D ) are invertible is also connected to the study of robust sta-bilization in control engineering, see [5, 23].Section 4 addresses (Q2) and shows how to move between realiza-tions on different canonical domains while preserving delicate regularitybehavior; see Theorems 4.1 and 4.2. Specifically, we show that on thelevel of algebra, the set of definition of a realization is the same as thatwhen the domains have been conformally transformed, excepting ob-vious obstructions. In the noncommutative case, the results we obtainare completely clean, “minimal” realization formulae that are canoni-cal and therefore have maximal domain, similar to the results in [49]. KELLY BICKEL, J. E. PASCOE, AND RYAN TULLY-DOYLE
Section 5 contains an application of these theorems; we use the canon-ical realization from Theorem 2.1 for inner Schur functions on D toobtain canonical representations for real Pick functions on Π . Section 6 addresses our progress on the McCarthy Champagne con-jecture. We first combine the machinery from Section 4 with Theorem3.2 to establish
Theorem. f arises from a two-variable quasi-rational function φ , then the MCC holds for f .For the exact details of the statement, including the domain where f is globally matrix monotone as well as the connection between f and its associated quasi-rational function φ , see Section 6. In that sec-tion, we also study a class of d -variable Pick-Agler functions known as commutative perspective functions , which appear in the operatormeans literature [37, 25, 24, 26]. We show that the noncommutativeL¨owner theorem from [50] implies that Theorem. f is a d -variable commutative perspective function,then the MCC holds for f .One surprising aspect of the precise statement of Theorem 6.2 isthe following: it only assumes that f is locally matrix monotone ona positive cone C ⊆ (0 , ∞ ) d but concludes that f must actually beglobally matrix monotone on all of (0 , ∞ ) d .2. Two-variable realization formulae
We begin with the technical setup for the de Branges-Rovnyak canon-ical model theory for two variable inner functions from [20, 21]. Through-out this section, let φ : D → D be a two variable inner function.Denote by H = H ( D ) the Hardy space on the bidisk. First, werecord some useful facts about the action of multiplication operatorson H . For j = 1 , , let M z j denote multiplication by z j in H and ONCRETE REALIZATIONS AND THE MCC 9 recall that the adjoints are the backward shift operators defined by( M ∗ z f )( z ) = f ( z ) − f ( z , z , ( M ∗ z f )( z ) = f ( z ) − f (0 , z ) z for all f ∈ H and so we have(2.1) f ( z ) = z ( M ∗ z f )( z ) + f ( z , , (2.2) f ( z ) = z ( M ∗ z f )( z ) + f (0 , z ) . Evaluating (2.2) at z = 0 gives(2.3) f ( z ,
0) = z ( M ∗ z f )( z ,
0) + f (0 , , which can be plugged into (2.1) to produce the formula(2.4) f ( z ) = z ( M ∗ z f )( z ) + z ( M ∗ z f )( z ,
0) + f (0 , . We now define the enveloping reproducing kernel Hilbert space for φ and the structured subspaces upon which the Agler model equationwill be built. • Let K φ be the reproducing kernel Hilbert space K φ = H " − φ ( z ) φ ( w )(1 − z w )(1 − z w ) = H ⊖ φH ; • S max1 = the maximum subspace of K φ invariant under M z ; • S min2 = K φ ⊖ S max1 ;Here H ( K ) denotes the Hilbert space of functions with reproducingkernel K . In [17], Ball, Sadosky, and Vinnikov showed that with thesedefinitions, S min2 is invariant under M z . We can then define these keyHilbert spaces: • H ( K max1 ) = S max1 ⊖ z S max1 ; • H ( K min2 ) = S min2 ⊖ z S min2 ,where z j is shorthand for M z j . As K φ = S max1 ⊕ S min2 , their reproducingkernels satisfy the question1 − φ ( z ) φ ( w )(1 − z w )(1 − z w ) = K max1 ( z, w )1 − z w + K min2 ( z, w )1 − z w . This immediately gives the associated
Agler model equation (2.5) 1 − φ ( z ) φ ( w ) = (1 − z w ) K min2 ( z, w ) + (1 − z w ) K max1 ( z, w ) . Set H φ = H ( K min2 ) ⊕ H ( K max1 ) , so that each f ∈ H φ can be written uniquely as f = f + f for f ∈ H ( K min2 ) , f ∈ H ( K max1 ). Note that in contrast to Agler’s originalapproach in [2], here we have explicit kernel structures to work with,which will allow direct calculations involving functions in H φ .We now use (2.5) to derive a realization formula for φ . Define anoperator V so that for all w ∈ D , V w k min2 ,w w k max1 ,w φ ( w ) k min2 ,w k max1 ,w , where k max1 ,w = K max1 ( · , w ) and k min2 ,w = K min2 ( · , w ). Then the argumentsin [21, pp. 6316-6318] imply that V extends to a unique isometryon C ⊕ H ( K min2 ) ⊕ H ( K max1 ). More generally, this type of argument isknown as a lurking isometry argument, see [3], but often the underlyingHilbert space needs to be enlarged to guarantee that the resulting V isisometric. Now, if we write(2.6) V ∗ = " A BC D : " C H φ " C H φ , then for all z ∈ D ,(2.7) φ ( z ) = A + B (1 − E z D ) − E z C, where E z = M z P + M z P and P , P ∈ L ( H φ ) are defined as fol-lows: P is the projection onto H ( K min2 ), and P is the projection onto H ( K max1 ).We can take advantage of the explicit structure of the model setupto derive concrete formulae for the blocks of V ∗ . ONCRETE REALIZATIONS AND THE MCC 11
Theorem 2.1.
Let φ be a two-variable inner function with concreterealization (2.7) . Then the following formulas hold: (1) For all x ∈ C , A is given by Ax = φ (0) x. (2) For all f ∈ H φ , B is given by B " f f = ( f + f )(0) . (3) For all x ∈ C , C is given by Cx = " P M ∗ z φP M ∗ z φ x. (4) For all f ∈ H φ , D is given by D " f f = " P M ∗ z ( f + f ) P M ∗ z ( f + f ) . Proof.
The formulas for A and B are given in [20, Remark 5 . C and D are proved in Lemmas 2.4 and 2.5 below. (cid:3) Remark . The more general class of so-called weakly coisometricrealizations for d -variable Schur-Agler functions (Schur functions thatalso satisfy von Neumann’s inequality) and their associated A, B, C, D formulas were studied earlier in [14]. Specifically, Definition 3 . . A and B above andimply that C and D must each satisfy a so-called structured Gleasonproblem.The concrete function theory interpretations for C and D in Theorem2.1 are also related to the technical and extensive work in [15]. Inparticular, in Theorem 5 .
9, the authors assume that a given d -variableSchur-Agler function ϕ possesses a so-called minimal augmented Aglerdecomposition and use it to construct a specific unitary realization for ϕ via the theory of scattering systems and formal reproducing kernel Hilbert spaces. The
A, B, C, D formulas that they obtain are quitesimilar to those of the cosimetric realization in Theorem 2.1 above.The following lemma will simplify later computations. Part of itappears as Proposition 3 . Lemma 2.3.
Let φ be a two-variable inner function with associatedHilbert spaces defined as above. Then M ∗ z φ ∈ S max1 and M ∗ z φ ∈ S min2 .Furthermore M ∗ z H φ ⊆ S max1 and M ∗ z H φ ⊆ S min2 . Proof. As S min2 is invariant under M z , it follows easily that S max1 isinvariant under M ∗ z . Thus, M ∗ z H ( K max1 ) ⊆ S max1 . Now rewrite themodel equation (2.5) as the following equality of positive kernels:11 − z w + z w K max1 ( z, w )1 − z w = φ ( z ) φ ( w )1 − z w + K min2 ( z, w ) + K max1 ( z, w )1 − z w . This shows that φ and each f ∈ H ( K min2 ) can be written as g ( z ) + z h ( z ) where g ∈ H ( D ) and h ∈ S max1 . Then the definition of M ∗ z immediately implies M ∗ z φ ∈ S max1 and M ∗ z H ( K min2 ) ⊆ S max1 , whichestablishes the S max1 inclusions. The S min2 inclusions follow from ananalogous argument. (cid:3) We can now establish the formulae for C and D . Lemma 2.4.
Let φ be a two-variable inner function with concrete re-alization (2.7) . Thenfor all f = " f f ∈ H φ , D " f f = " P M ∗ z ( f + f ) P M ∗ z ( f + f ) . Proof.
Make the decomposition Df = " ( Df ) ( Df ) . We first establish theformula for ( Df ) and then consider ( Df ) . By [21, Remark 5.6], ( Df ) is the unique function in H ( K max1 ) sat-isfying( Df )(0 , z ) = ( f + f )(0 , z ) − ( f + f )(0) z = M ∗ z ( f + f )(0 , z ) . ONCRETE REALIZATIONS AND THE MCC 13
By Lemma 2.3, we have M ∗ z ( f + f ) ∈ S max1 . Thus, we can write M ∗ z ( f + f ) = P M ∗ z ( f + f ) + (1 − P ) M ∗ z ( f + f ) , where (1 − P ) projects S max1 onto z S max1 . As (1 − P ) M ∗ z ( f + f ) isthus divisible by z , we have M ∗ z ( f + f )(0 , z ) = P M ∗ z ( f + f )(0 , z )and so by uniqueness, ( Df ) = P M ∗ z ( f + f ) . To establish the formula for ( Df ) , note that by [21, Remark 5.6],( Df ) is the unique function in H ( K min2 ) satisfying(2.8) ( Df ) ( z ) = ( f + f )( z ) − ( f + f )(0) − z ( Df ) ( z ) z . By the formula in (2.4), we have( f + f )( z ) = z M ∗ z ( f + f )( z ) + z M ∗ z ( f + f )( z ,
0) + ( f + f )(0) , and so( Df ) ( z ) = f ( z ) + f ( z ) − f (0) − f (0) − z ( Df ) ( z ) z = z M ∗ z ( f + f )( z ) + z M ∗ z ( f + f )( z , − z P M ∗ z ( f + f )( z ) z = M ∗ z ( f + f )( z ,
0) + z (1 − P ) M ∗ z ( f + f )( z ) z = M ∗ z ( f + f )( z ,
0) + z M ∗ z (1 − P ) M ∗ z ( f + f )( z )= M ∗ z (cid:2) ( f + f )( z ,
0) + z (1 − P ) M ∗ z ( f + f )( z ) (cid:3) = M ∗ z (cid:2) ( f + f )( z ,
0) + z M ∗ z ( f + f )( z ) − z P M ∗ z ( f + f )( z ) (cid:3) = M ∗ z (cid:2) ( f + f )( z ) − z P M ∗ z ( f + f )( z ) (cid:3) (by (2.1))= M ∗ z (1 − z P M ∗ z )( f + f )( z ) , where we again used the fact that (1 − P ) M ∗ z ( f + f ) is divisible by z . So, we have( Df ) = M ∗ z (1 − M z P M ∗ z )( f + f )= P M ∗ z ( f + f ) + P M z M ∗ z P M ∗ z ( f + f ) , since ( Df ) ∈ H ( K min2 ). By Lemma 2.3, M ∗ z P M ∗ z ( f + f ) ∈ S min2 .Thus, M z M ∗ z P M ∗ z ( f + f ) ∈ z S min2 and so, is annihilated by P .This implies ( Df ) = P M ∗ z ( f + f ) , and establishes the claim. (cid:3) Lemma 2.5.
Let φ be a two-variable inner function with concrete re-alization (2.7) . For all x ∈ C , C is given by Cx = " P M ∗ z φP M ∗ z φ x. Proof.
The proof is similar to the argument for D in Lemma 2.4, sowe give a sketch of the idea but omit some of the finer details. Bylinearity, we can let x = 1. By [21, Remark 5.6], ( C is the uniquefunction in H ( K max1 ) with( C (0 , z ) = ( M ∗ z φ )(0 , z ) . Lemma 2.3, M ∗ z φ ∈ S max1 and then the same rationale as in Lemma2.4 implies that ( C = P M ∗ z φ. To handle ( C , write φ ( z ) = z M ∗ z φ ( z ) + z M ∗ z φ ( z ,
0) + φ (0) . Then [21, Remark 5.6] implies that ( C satisfies(2.9) z ( C ( z ) + z ( C ( z ) = φ ( z ) − φ (0) . Substituting the formulas for φ and ( C into (2.9) and solving for( C yields( C ( z ) = M ∗ z φ ( z ,
0) + z M ∗ z φ ( z ) − z P M ∗ z φ ( z ) z = M ∗ z φ ( z ,
0) + M ∗ z (cid:0) z M ∗ z φ − z P M ∗ z φ (cid:1) ( z )= M ∗ z φ ( z ) − z M ∗ z P M ∗ z φ ( z ) , where we used the fact that M ∗ z φ ∈ S max1 . As ( C ∈ H ( K min2 ),( C = P M ∗ z φ − P M z M ∗ z P M ∗ z φ. Then Lemma 2.3 implies that M ∗ z P M ∗ z φ ∈ S min2 and so, M z M ∗ z P M ∗ z φ is annihilated by P . This implies ( C = P M ∗ z φ and completes theproof. (cid:3) ONCRETE REALIZATIONS AND THE MCC 15
We now show that the operator D exhibits additional behavior re-sembling that of the backward shift M ∗ z i . Proposition 2.6.
Let φ be a two-variable inner function with concreterealization (2.7) . Then for all w ∈ D D " k min2 ,w k max1 ,w = " w k min2 ,w w k max1 ,w − φ ( w ) F, where F = " P M ∗ z φP M ∗ z φ . Proof.
By Lemma 2.3, M ∗ z ( k max1 ,w + k min2 ,w ) ∈ S max1 . Rearranging the modelequation (2.5) and applying the operator P M ∗ z to each side gives P M ∗ z [ k max1 ,w + k min2 ,w ] = P [ z w M ∗ z k min2 ,w + w k max1 ,w − φ ( w ) M ∗ z φ ]= w k max1 ,w − φ ( w ) P M ∗ z φ, (2.10)since z M ∗ z k min2 ,w ∈ z S max1 and hence, is annihilated by P . Similarly, P M ∗ z [ k max1 ,w + k min2 ,w ] = P [ w k min2 ,w + z w M ∗ z k max1 ,w − φ ( w ) M ∗ z φ ]= w k min2 ,w − φ ( w ) P M ∗ z φ, (2.11)since z M ∗ z k max1 ,w ∈ z S min2 . Now applying Lemma 2.4 to D " k min2 ,w k max1 ,w andusing the expressions in (2.10), (2.11) gives the desired formula. (cid:3) Boundary behavior of quasi-rational functions
As in the last section, let φ be a two-variable inner function on D .We now examine the behavior of the concrete realization of φ from(2.7) at points on the distinguished boundary T . The goal is to showthat if φ extends continuously at part of the boundary, then so does therealization. Equivalently, we want to show that the operator 1 − E τ D isinvertible on some open set of boundary points where φ is well behaved.This problem is generally intractable via current methods, so werestrict to a special class of inner functions. Specifically, we say that aSchur function φ is quasi-rational with respect to an open I ⊆ T if φ isinner and extends continuously to T × I with | φ ( τ ) | = 1 for τ ∈ T × I .To get a sense of the definition, recall that every one-variable inner function that extends continuously to T is a finite Blaschke product.Thus, if φ is quasi-rational, then for each τ ∈ I , the one-variablefunction φ ( · , τ ) must be a finite Blaschke product. Remark . Before proceeding further, one should note that the setof quasi-rational functions is quite large. To generate examples, let p ∈ C [ z , z ] be a polynomial of degree ( m , m ) that does not vanishon D and let ψ = ˜ pp , where ˜ p ( z ) = z m z m p (1 / ¯ z , / ¯ z ). Without lossof generality, we can assume that p, ˜ p have no common factors. Then ψ is a rational inner function (and all rational inner functions have thisform, see [57]) and we can define the set J ψ = { τ ∈ T : there exists τ ∈ T with p ( τ , τ ) = 0 } , which contains at most m m points. Now let θ be any one-variableinner function that extends continuously to an open set J ⊆ T . Let I ⊆ J be any open set such that θ ( I ) ⊆ T \ J ψ . Then the two-variablefunction φ defined by φ ( z ) = ψ ( z , ϕ ( z )) , is quasi-rational with respect to I . Furthermore, it is immediate thatthe set of quasi-rational functions with respect to I is closed underfinite products.The class of quasi-rational functions with respect to I is also closedin a stronger sense. Specifically, assume that ( φ n ) is a sequence ofquasi-rational functions on I that converges to some function φ bothin the H ( D ) norm and locally uniformly on D ∪ ( T × I ). The firstcondition implies that the limit function φ is inner and the secondcondition implies that φ extends continuously to T × I . Thus, φ isquasi-rational with respect to I .Then for quasi-rational functions, we prove the following result: Theorem 3.2.
Let φ be quasi-rational with respect to an open I ⊆ T .Then in the concrete realization (2.7) , the operator − E τ D is invertiblefor all τ ∈ T × I . ONCRETE REALIZATIONS AND THE MCC 17
Before proving the theorem, we prepare some model machinery.
Observation 3.3.
By Theorem 1.5 in [20], there is an open set Ωcontaining ( D × I ) ∪ ( T × D ) ∪ D on which φ and all functions in H ( K max1 )and H ( K min2 ) extend to be analytic. Furthermore, point evaluation inΩ is bounded on these spaces, and the kernels K max1 , K min2 extend tobe sesquianalytic on Ω × Ω.Observe that for τ ∈ I , the one-variable inner function φ τ = φ ( · , τ )is well defined and possesses an associated one-variable reproducingkernel Hilbert space defined by K φ τ = H " − φ τ ( z ) φ τ ( w )1 − z w , which is a subspace of the one-variable Hardy space H ( D ). We con-nect these K φ τ to the subspaces associated to our realizations via thefollowing lemma. Lemma 3.4.
Let φ be quasi-rational with respect to an open I ⊆ T .Then the map J τ : H ( K min2 ) → K φ τ defined by J τ f = f ( · , τ ) isunitary for all τ ∈ I .Proof. This proof uses ideas from the proofs of [20, Theorem 1.6] and[19, Theorem 2.2]. For this proof, one should recall that H ( K min2 ) is asubspace of H ( D ) and K φ τ is a subspace of H ( D ).We first claim that for τ ∈ I , the restriction map J τ : H ( K min2 ) → H ( D ) preserves inner products. Fix functions f, g ∈ H ( K min2 ). Thenfor almost every z ∈ T , f ( · , z ) , g ( · , z ) ∈ L ( T ) and we can define(3.1) F f,g ( z ) = h f ( · , z ) , g ( · , z ) i L ( T ) . Let σ denote normalized Lebesgue measure. Then by H¨older’s inequal-ity, we obtain Z T | F f,g ( z ) | dσ ( z ) ≤ Z T k f ( · , z ) k L ( T ) k g ( · , z ) k L ( T ) dσ ( z ) ≤ k f k H k g k H , which implies F f,g ∈ L ( T ). Since f, g ∈ H ( K min2 ) and H ( K min2 ) ⊥ H z H ( K min2 ), we have f ⊥ L z j g for all j ∈ Z / { } . Then the Fourier coefficients of F f,g for j ∈ Z / { } are given by d F f,g ( − j ) = Z T z j F f,g ( z ) dσ ( z ) = Z T z j f ( z ) g ( z ) dσ ( z ) = 0 , and so it is straightforward that F f,g ( z ) = d F f,g (0) = h f, g i H ( K min2 ) for a.e. z ∈ T . By Observation 3.3, f and g are analytic on an open set Ω containing D × I , which implies both that for every τ ∈ I , f ( · , τ ) , g ( · , τ ) ∈ H ( D )and the formula for F f,g in (3.1) is well defined and continuous on I .This immediately gives h f ( · , τ ) , g ( · , τ ) i H ( D ) = F f,g ( τ ) = d F f,g (0) = h f, g i H ( K min2 ) , so the restriction map preserves inner products for each τ ∈ I .To finish the proof, we need to show J τ maps onto K φ τ . By Obser-vation 3.3, for any τ ∈ I , we can let z , w → τ in the model equation(2.5) to obtain1 − φ τ ( z ) φ τ ( w )1 − z w = K min2 (( z , τ ) , ( w , τ )) = J τ (cid:0) k min2 , ( w ,τ ) (cid:1) ( z ) . To show the range of J τ is in K φ τ , assume that f ∈ H ( K min2 ) and J τ f ⊥ K φ τ in H ( D ). Since J τ preserves inner products, this impliesthat for all w ∈ D (cid:10) f ( · , τ ) , k min2 , ( w ,τ ) ( · , τ ) (cid:11) H ( D ) = (cid:10) f, k min2 , ( w ,τ ) (cid:11) H ( K min2 ) = f ( w , τ ) . Since J τ preserves norms, f ≡ J τ maps into K φ τ .Finally, as J τ preserves norms and its range contains all of the re-producing kernel functions of K φ τ , J τ must be surjective. (cid:3) We are now ready to prove the main theorem.
ONCRETE REALIZATIONS AND THE MCC 19
Proof of Theorem 3.2.
Fix ( τ , τ ) ∈ T × I. The proof has two parts.We start by showing that the operator (1 − E τ D ) has dense range andthen we will show (1 − E τ D ) is bounded below.First, proceed by contradiction and assume that (1 − E τ D ) does nothave dense range. Recall that Proposition 2.6 implies(3.2) (1 − E τ D ) " k min2 ,w k max1 ,w = " (1 − τ w ) k min2 ,w (1 − τ w ) k max1 ,w + φ ( w ) E τ F, where F is defined in Proposition 2.6. Then there must exist somenon-trivial g ∈ H φ orthogonal to all functions with the form given in(3.2). Writing g = " g g for g ∈ H ( K min2 ) , and g ∈ H ( K max1 ) , we can compute0 = * g, (1 − E τ D ) " k min2 ,w k max1 ,w H φ = (1 − τ w ) g ( w ) + (1 − τ w ) g ( w ) + φ ( w ) h g, E τ F i H φ , for all w ∈ D . By Observation 3.3, we can take w → τ and reducethis to 0 = φ ( τ ) h g, E τ F i H φ and so h g, E τ F i H φ = 0. Then(3.3) (1 − τ w ) g ( w ) = − (1 − τ w ) g ( w )and in particular, by Observation 3.3, we can take limits to points in D × I to conclude g ( z , τ ) = 0 for all z ∈ D . Since g ∈ H ( K min2 ), anapplication of Lemma 3.4 implies that k g k H ( K min2 ) = k g ( · , τ ) k K φτ = 0 . Thus g ≡
0, and by (3.3) we also have g ≡
0. Then g ≡
0, which is acontradiction. We conclude that (1 − E τ D ) has dense range.Now we show that (1 − E τ D ) is bounded below. Proceeding bycontradiction, assume that (1 − E τ D ) is not bounded below. Then there is a sequence of functions { g n } ⊂ H φ such that k g n k H φ = 1 andlim n →∞ k (1 − E τ D ) g n k H φ = 0.Let ˜ I be a closed interval in I containing τ in its interior. By Ob-servation 3.3 and the uniform boundedness principle, point evaluationin H φ is uniformly bounded on K := D × ˜ I . That is, there is a C > | f ( z ) | ≤ C k f k H φ for all z ∈ K, f ∈ H φ . Furthermore, for i = 1 ,
2, this implies that for z ∈ K , | z i ( Dg n ) i ( z ) − τ i z i g ni ( z ) | = | z i (1 − E τ Dg n ) i ( z ) | ≤ C k (1 − E τ Dg n ) k H φ . By (2.8), we can conclude that | g n ( z ) − τ z g n ( z ) + g n ( z ) − τ z g n ( z ) − g n (0) − g n (0) | = | z ( Dg n ) ( z ) − τ z g n ( z ) + z ( Dg n ) ( z ) − τ z g n ( z ) |≤ C k (1 − E τ D ) g n k H φ for all z ∈ K. Setting z = τ , we have | g n (0) + g n (0) | ≤ C k (1 − E τ D ) g n k H φ and so | (1 − τ z ) g n ( z ) + (1 − τ z ) g n ( z ) | ≤ C k (1 − E τ D ) g n k H φ , for all z ∈ K. In particular, setting z = τ gives(3.5) | (1 − τ z ) g n ( z , τ ) | ≤ C k (1 − E τ D ) g n k H φ , for all z ∈ D . We can use this to deduce that k g n k H φ →
0. First, fixa small ǫ > σ denote normalized Lebesgue measure on T (or T , depending on the context) choose a compact interval K ⊆ T centered at τ such that σ ( K ) = ǫ . Then dist( T \ K , τ ) = ǫ/ . Then
ONCRETE REALIZATIONS AND THE MCC 21 by Lemma 3.4 and equations (3.4), (3.5), we have k g n k H φ = k g n ( · , τ ) k H = Z K | g n ( z , τ ) | dσ ( z ) + Z T \K | (1 − τ z ) g n ( z , τ ) | | z − τ | dσ ( z ) ≤ σ ( K ) C k g n k H φ + 16 C dist( T \ K , τ ) k (1 − E τ D ) g n k H φ ≤ ǫC + 64 C ǫ k (1 − E τ D ) g n k H φ . Choose N such that for all n ≥ N , the latter term is less than ǫ. Thisshows k g n k H φ → . Now, consider g n . By our original assumptions and the fact that k g n k H φ → , we can conclude that k g n k H φ → k (1 − E τ D ) g n k H φ → . Examining the first component in the second limit yields k τ ( Dg n ) k H φ → , and so k M z ( Dg n ) k H → k g n − τ ( Dg n ) k H φ → , and so k M z ( τ g n − ( Dg n ) ) k H → . Thus (2.8) implies that k (1 − M z τ ) g n − g n (0) k H = k M z ( Dg n ) + M z (( Dg n − τ g n ) ) k H → . From earlier in the argument, we know that | g n (0) + g n (0) | → k g n k H →
0. This implies that g n (0) → k (1 − M z τ ) g n k H → . We claim that this implies k g n k H φ → . To see this, fix a small ǫ >
K ⊆ ˜ I a compact interval centered at τ with σ ( K ) = ǫ . Notethat such a K exists for ǫ sufficiently small. Then dist( T \ K , τ ) = ǫ/ . Then by (3.4) and (3.6), k g n k H φ = Z T ×K | g n ( z ) | dσ ( z ) + Z T × ( T \K ) | (1 − τ z ) g n ( z ) | | z − τ | dσ ( z ) ≤ σ ( K ) C k g n k H φ + 1dist( T \ K , τ ) k (1 − z τ ) g n k H ≤ ǫC + 4 ǫ k (1 − M z τ ) g n k H . Choose N such that for all n ≥ N , the latter term is less than ǫ. Thisshows k g n k H φ → , a contradiction, which completes the proof. (cid:3) It seems plausible that Theorem 3.2 should hold if φ is inner andextends continuously to I × I for open sets I , I ⊆ T . While we havenot been able to prove this, we can show that the (1 − E τ D ) operatorshave dense range. Proposition 3.5.
Let φ be an inner function on D and assume φ extends continuously to I × I for open sets I , I ⊆ T . Then, for each τ ∈ I × I , the operator (1 − E τ D ) has dense range.Proof. Fix ( τ , τ ) ∈ I × I . As in the proof of Theorem 3.2, assumethat g ∈ H φ is orthogonal to the range of 1 − E τ D . Write g = " g g for g ∈ H ( K min2 ) , and g ∈ H ( K max1 ) . Then one can basically follow the proof of Theorem 3.2, but directlyapply Theorem 1 . − τ w ) g ( w ) = − (1 − τ w ) g ( w )for all w ∈ Ω, where Ω is an open set containing D ∪ ( I × D ) ∪ ( D × I ) ∪ ( I × I ) and all elements of H φ extend to be holomorphic. on Ω.For w ∈ Ω wherever the expression makes sense, define a function f by f ( w ) = g ( w )1 − ¯ τ w = − g ( w )1 − ¯ τ w . The first formula says f is holomorphic on Ω \ { ( w , w ) : w = τ } , andthe second formula says f is holomorphic on Ω \ { ( w , w ) : w = τ } . ONCRETE REALIZATIONS AND THE MCC 23
This implies that f is holomorphic on Ω \ { τ } . Holomorphic functionson open sets in C cannot have isolated singularities and so, f mustbe holomorphic on Ω . To show that f ∈ H ( D ), choose compact sets K , K ⊆ T containing τ and τ respectively such that K ×K ⊂ I × I and dist( T \ K j , τ j ) > j = 1 ,
2. Then since K × K ⊂ Ω, f isbounded on K × K and we have: k f k H = Z K ×K | f ( z ) | dσ ( z ) + Z T × ( T \K ) (cid:12)(cid:12)(cid:12)(cid:12) g ( z )1 − ¯ τ z (cid:12)(cid:12)(cid:12)(cid:12) dσ ( z )+ Z ( T \K ) × T (cid:12)(cid:12)(cid:12)(cid:12) g ( z )1 − ¯ τ z (cid:12)(cid:12)(cid:12)(cid:12) dσ ( z ) < ∞ . Furthermore, observe that for each N ∈ N , f ( w ) = g ( w )1 − ¯ τ w = N − X n =0 g ( w )¯ τ n w n + ¯ τ N w N g ( w )1 − ¯ τ w . Since g ∈ H ( K min2 ), we know g ⊥ H w n g for all n > k f k H ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N − X n =0 g ( w )¯ τ n w n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H − (cid:13)(cid:13)(cid:13)(cid:13) ¯ τ N w N g ( w )1 − ¯ τ w (cid:13)(cid:13)(cid:13)(cid:13) H = N − X n =0 k w n g ( w ) k H ! / − k f k H = √ N k g k H − k f k H . Since this holds for all N , it follows that g ≡ g ≡
0, whichproves the claim. (cid:3)
However, as discussed in the following remark, the proof showingthat the (1 − E τ D ) operators are bounded below does not translate tothis setting. Remark . Numerous times, the proof of Theorem 3.2 uses the uni-form boundedness of point evaluations delineated in (3.4). For example,this is used to deduce that(3.7) Z T ×K | g n ( z ) | dσ ( z ) . σ ( K ) for K ⊆ I a small set containing τ .It is not clear how to obtain such bounds if φ only extends contin-uously to a more general product set I × I . To see how we mightobtain this inequality using other means, recall that point evaluationson D × I are bounded on H φ . Then Z T ×K | g n ( z ) | dσ ( z ) = Z T ×K lim r ր (cid:12)(cid:12)(cid:12)(cid:10) g n , k max1 , ( rz ,z ) (cid:11) H (cid:12)(cid:12)(cid:12) dσ ( z ) ≤ k g n k H Z T ×K lim r ր (cid:13)(cid:13) k max1 , ( rz ,z ) (cid:13)(cid:13) H dσ ( z ) . Then (3.7) would follow if Z T ×K lim r ր (cid:13)(cid:13) k max1 , ( rz ,z ) (cid:13)(cid:13) H dσ ( z ) . σ ( K ) . But, a straightforward computation using the model equation (2.5) andthe one-variable Julia-Carath´eodory theorem giveslim r ր (cid:13)(cid:13) k max1 , ( rz ,z ) (cid:13)(cid:13) H = lim r ր − | φ ( rz , z ) | − r ≈ | ∂z φ ( z , z ) | , where ∂z φ ( z , z ) is the non-tangential derivative of φ ( · , z ) at z ,which is defined as long as k k max1 , ( rz ,z ) k H is bounded as r ր
1. Thenthe desired equality becomes Z T ×K lim r ր (cid:13)(cid:13) k max1 , ( rz ,z ) (cid:13)(cid:13) H dσ ( z ) ≈ Z T ×K | ∂z φ ( z , z ) | dσ ( z ) . σ ( K ) . This uniform H derivative bound certainly forces φ ( · , z ) to be a finiteBlaschke product for a.e. z ∈ I and likely imposes even more stringentregularity conditions on φ . Therefore, new techniques would be neededto show that the (1 − E τ D ) operators are bounded below for φ thatpossess weaker regularity than that assumed in Theorem 3.2.4. Some algebraic identities
In this section, we collect some algebraic identities satisfied by gen-eral noncommutative indeterminants which will allow us to convertbetween various representation formulae.
ONCRETE REALIZATIONS AND THE MCC 25
Algebraic realizations of noncommutative Schur-type functions arecalled Fornasini-Marchesini realizations, after the pioneering work in[27]. We introduce a new algebraic version of the usual noncommu-tative Herglotz realization (as in [55]), the so-called Herglotz-Nouveauformula. Finally, we refer to the noncommutative Nevanlinna realiza-tion [48, 46, 49].4.1.
The block by matrix inverse formula. In what follows,we will need formulas for inverses of block 2 × X be the following 2 × X = " Q RS V . Provided that certain related operators are invertible, this partitionyields useful formulas for X − . For example, if Q and V − SQ − R areinvertible, then so is X and X − is given by " Q − + Q − R ( V − SQ − R ) − SQ − − Q − R ( V − SQ − R ) − − ( V − SQ − R ) − SQ − ( V − SQ − R ) − . Similarly, if V and Q − RV − S are invertible, then X − exists and isgiven by the formula " ( Q − RV − S ) − − ( Q − RV − S ) − RV − − V − S ( Q − RV − S ) − V − + V − S ( Q − RV − S ) − RV − . See for example, [32, p. 18].4.2.
Between Fornasini-Marchesini and Herglotz-Nouveau for-mulae.Theorem 4.1.
Let
A, B, C, D, Z be operators taking various Hilbertspaces to other various Hilbert spaces such that the expression
Φ = A + B (1 − ZD ) − ZC is well defined and − Φ is invertible. Let Θ = 1 + Φ1 − Φ . Then the expression − " A BZC ZD is an invertible operator and (4.1) Θ = " ∗ − " A BZC ZD − " A BZC ZD . Proof.
By hypothesis, 1 − ZD is invertible and (1 − Φ) − , which is theSchur complement of 1 − ZD in " − A − B − ZC − ZD , exists as 1 − Φis invertible by assumption. This implies that 1 − " A BZC ZD is aninvertible operator.Applying the inverse formula for a block two by two matrix, we get " − A − B − ZC − ZD − = " S SB (1 − ZD ) − ∗ ∗ where(4.2) S = (1 − A − B (1 − DZ ) − ZC ) − = (1 − Φ) − , and ∗ denotes some quantity that will be immaterial to our calculation.On substitution into 4.1, we get " ∗ − " A BZC ZD − " A BZC ZD = " ∗ " S SB (1 − ZD ) − ∗ ∗ A + 1 BZC ZD + 1 = h S SB (1 − ZD ) − i " AZC = S (1 + A ) + SB (1 − ZD ) − ZC = S (1 + A + B (1 − ZD ) − ZC ) ONCRETE REALIZATIONS AND THE MCC 27 = 1 + Φ1 − Φ , which proves the claim. (cid:3) Between Herglotz-Nouveau and Nevanlinna formulae.Theorem 4.2.
Let U = " A BC D be a block operator such that − U is invertible. Let Z be an operator such that − Z is invertible and (4.3) Θ = " ∗ − " A BZC ZD − " A BZC ZD is well defined. Let W = i Z − Z so Z = W − iW + i , let T = i (1 + U )(1 − U ) − = " T T T T , and let Ψ = i Θ . Then the expression ( W + T ) is an invertible operatorand Ψ = T − T ( W + T ) − T . Proof.
For ease of notation, set α = " . The formula for Θ givesΨ = i Θ = iα ∗ − " W − iW + i A BC D − " W − iW + i A BC D α = iα ∗ " W + i − " W − i U ! − " W + i + " W − i U ! α = iα ∗ " W (1 − U ) + " i (1 + U ) ! − " W (1 + U ) + " i (1 − U ) ! α. Recalling that T = i (1 + U )(1 − U ) − , we haveΨ = α ∗ (1 − U ) − " W + " T ! − " W T + " − (1 − U ) α. The expression " W + " T ! − is the conjugation of a welldefined expression from the original equation by invertible operators,and thus remains well defined. Then note that " W + " T ! − = " T W + T − which implies that the expression W + T is invertible.Now, writing " W T + " − = " W + " T ! T − " (1 + T ) , givesΨ = α ∗ (1 − U ) − T (1 − U ) α − α ∗ (1 − U ) − " W + " T ! − " (1 + T )(1 − U ) α = α ∗ T α − α ∗ (1 − U ) − " W + " T ! − " (1 + T )(1 − U ) α. Observe that(1 + T )(1 − U ) = ((1 − U ) − (1 + U ) )(1 − U ) − = − U (1 − U ) − , and by the inversion formula for 2 × T + W are both invertible, we have " W + " T ! − " = " T W + T − " = " − ( W + T ) − T ( W + T ) − = " W + T ) − = " ( W + T ) − h i . ONCRETE REALIZATIONS AND THE MCC 29
Then, using the definition of α , the equation simplifies toΨ = T + 4 h i (1 − U ) − " ( W + T ) − h i U (1 − U ) − " . Further, observe that T = i (1 − U + 2 U )(1 − U ) − = i + 2 iU (1 − U ) − , so 2 U (1 − U ) − = − iT − ,T = i ( U − − U ) − = − i + 2 i (1 − U ) − , so 2(1 − U ) − = − iT + 1 . Those formulas imply that2 h i (1 − U ) − " = − iT and 2 h i U (1 − U ) − " = − iT and so, the formula for Ψ becomesΨ = T − T ( W + T ) − T , which is what we were trying to show. (cid:3) Concrete Nevanlinna formulae
We can use Theorem 4.2 to translate the concrete realizations forinner functions on D from Section 2 to realizations for Pick functionson Π .First, assume that φ is an inner function on D . Then φ has a modelrepresentation as in (2.7), where the realization operator U := " A BC D : " C H φ " C H φ has block formulas given in Theorem 2.1. Define α : D → Π by α ( z ) = i z − z and α − : Π → D by α − ( w ) = w − iw + i . Then ψ = α ◦ φ ◦ α − is an inner Pick function on Π , where inner meansthat ψ is real valued for almost every point in R . Here, we shouldmention that the notation φ ◦ α − is short-hand for φ ◦ ( α − , α − ) andwill be used throughout the rest of the paper. If (1 − U ) is invertible, then Theorem 4.2 implies that if T = i (1 + U )(1 − U ) − = " T T T T on C ⊕ H φ , then ψ ( w ) = T − T ( E w + T ) − T for w ∈ Π . If φ has sufficient regularity at (1 , T , T , and T . Progress towardsa similar description of T is described in Remark 5.2. Theorem 5.1.
Let φ and ψ be as above. Assume that φ extends con-tinuously to a neighborhood of (1 , on T with φ (1 , = 1 and (2.7) extends to (1 , . Then by Theorem 1.5 in [20] , f (1 , exists for all f ∈ H φ and there is a k ∈ H φ such that f (1 ,
1) = h f, k i H φ for all f ∈ H φ . Then: i. For all x ∈ C , T is given by T x = i φ (1 , − φ (1 , x. ii. For all x ∈ C , T is given by T x = 2 iφ (1 , − φ (1 , k x. iii. For all f ∈ H φ , T is given by T f = 2 i − φ (1 , f (1 , . It follows from the proof that weaker regularity conditions are neededto obtain the formulas for T and T . Proof.
To obtain formulas for T , we require a formula for (1 − U ) − .Let c := (1 − φ (1 , − . Using the second formula in Section 4.1 andthe fact that φ (1 ,
1) = A + B (1 − D ) − C , one can obtain(1 − U ) − = " c c B (1 − D ) − c (1 − D ) − C (1 − D ) − + c (1 − D ) − CB (1 − D ) − . ONCRETE REALIZATIONS AND THE MCC 31
Using T = i (1+ U )(1 − U ) − , block matrix multiplication, and straight-forward simplification gives T = ic (1 + A + B (1 − D ) − C ) = i φ (1 , − φ (1 , ; T = ic ( C + (1 + D )(1 − D ) − C ) = i − φ (1 , (1 − D ) − C ; T = i (1 + A ) c B (1 − D ) − + iB (1 − D ) − + ic B (1 − D ) − CB (1 − D ) − = i − φ (1 , B (1 − D ) − ; T = i (1 + D )(1 − D ) − + c i (cid:0) D )(1 − D ) − (cid:1) CB (1 − D ) − = i (1 + D )(1 − D ) − + i − φ (1 , (1 − D ) − CB (1 − D ) − . The formula for T is immediate. To obtain the formula for T , set k w = k min2 ,w + k max1 ,w . Then by Proposition 2.6,(5.1) Dk w = E ¯ w k w − φ ( w ) F, where F is a function defined in Proposition 2.6 and by Theorem 2.1, Cx = F x for all x ∈ C . By Theorem 1.5 in [20], as w → (1 ,
1) with w ∈ D , k w → k weakly in H φ . One can use this to show Dk = k − φ (1 , F, which implies (1 − D ) − F = φ (1 , k . It follows immediately that for x ∈ C , T x = 2 ic (1 − D ) − Cx = 2 ic (1 − D ) − F x = iφ (1 , − φ (1 , k x. To study T and T , recall that we assumed φ continuously extendsto a neighborhood of (1 , − E ¯ w ) k w are dense in H φ . To see this, assume that g = g + g ∈ H φ and for all w ∈ D ,0 = h g, (1 − E ¯ w ) k w i H φ = (1 − w ) g ( w ) + (1 − w ) g ( w ) . Then the arguments in the proof of Proposition 3.5 imply that g ≡ . Thus it suffices to find a linear formula for T on functions of the form(1 − E ¯ w ) k w . To that end, note that (5.1) implies that(1 − E ¯ w ) k w = k w − Dk w − φ ( w ) F. Then by Theorem 2.1 and the formula (2.5), T (1 − E ¯ w ) k w = i − φ (1 , B (1 − D ) − (cid:16) k w − Dk w − φ ( w ) F (cid:17) = i − φ (1 , B ( k w − φ ( w ) φ (1 , k )= i − φ (1 , (cid:16) k w (0) − φ ( w ) φ (1 , k (0) (cid:17) = i − φ (1 , (cid:16) (1 − φ ( w ) φ (0 , − φ ( w ) φ (1 , − φ (1 , φ (0 , (cid:17) = i − φ (1 , (cid:16) − φ ( w ) φ (1 , (cid:17) = i − φ (1 , (1 − E ¯ w ) k w (1 , , which establishes the formula for T and completes the proof of thistheorem. Partial results concerning T are given in Remark 5.2. (cid:3) Remark . Recall from Theorem 5.1 that T = i (1 + D )(1 − D ) − + i − φ (1 , (1 − D ) − CB (1 − D ) − . The second piece of T combines the operators seen in T and T .Thus, we can combine our formulas for those two operators as follows:for all f ∈ H φ , i − φ (1 , (1 − D ) − CB (1 − D ) − f = i − φ (1 , (1 − D ) − Cf (1 ,
1) = if (1 , − φ (1 , k . We have not been able to deduce an explicit formula for the first pieceof T . However, we can compute it on the functions (1 − E ¯ w ) k w andso, under the regularity assumptions of Theorem 5.1, know its behavioron a dense set in H φ . Specifically, using Proposition 2.6, i (1 + D )(1 − D ) − (1 − E ¯ w ) k w = i (1 + D )( k w − φ ( w ) φ (1 , k )= i ( k w − φ ( w ) φ (1 , k + E ¯ w k w − φ ( w ) F − φ ( w ) φ (1 , k − φ (1 , F ))= i (1 + E ¯ w ) k w − iφ ( w ) φ (1 , k . ONCRETE REALIZATIONS AND THE MCC 33
We have not been able to find a bounded linear operator on H φ thatgives this formula on the functions (1 − E ¯ w ) k w and leave that as anopen question.6. The McCarthy Champagne conjecture
A large motivation for our present developments is the McCarthyChampagne Conjecture (MCC). A comprehensive discussion of theMCC was already provided in the introduction, but for the ease ofthe reader, let us recall the statement of the MCC here: (MCC): Every d -variable Pick-Agler function thatanalytically continues across an open convex set E ⊆ R d isglobally matrix monotone when restricted to E . Equivalently, this says that every locally matrix monotone functionon an open convex set E ⊆ R d is globally matrix monotone on E . Inthis section, we establish the MCC for two-variable Pick functions aris-ing from quasi-rational functions and for d -variable perspective func-tions.6.1. Quasi-rational functions.
As in Section 5, let α : D → Π de-note the Cayley transform given by α ( z ) = i (cid:0) z − z (cid:1) . Then as a directresult of Theorem 3.2 combined with Theorem 4.2, we can establishthe MCC for Pick functions arising from quasi-rational functions.
Theorem 6.1.
Let I ⊆ T be open, let φ be a nonconstant two-variablequasi-rational function on T × I , and define a Pick function f by f = α ◦ φ ◦ α − . Then f is globally matrix monotone on every open rectangle E ⊆ R × α ( I ) such that φ does not attain the value on α − ( E ) . Proof.
Let E ′ = J × J be a finite open rectangle with E ′ ⊆ E . Since E ′ is arbitrary, it suffices to show f is globally matrix monotone on E ′ .Let β = ( β , β ) be a pair of conformal self maps of Π such that β ( E ′ ) ⊆ (0 , ∞ ) and (0 , ∞ ) ∪ ( ∞ , ∞ ) ⊆ β ( E ) and furthermore f ( β − ( ∞ , ∞ )) ∈ R . Define F = f ◦ β − . Observe thateach β j is a one variable matrix monotone function. Thus, to show f isglobally matrix monotone on E ′ , we need only show that F is globallymatrix monotone on (0 , ∞ ) .To that end, observe that F = α ◦ Φ ◦ α − , where Φ = φ ◦ γ and γ = ( γ , γ ) is a pair of conformal self maps of D defined by γ j = α − ◦ β − j ◦ α . Then Φ is quasi-rational on T × I ′ , where I ′ = γ − ( I ). Tracingthrough the assumptions about φ , E , and β shows that (1 , ∈ T × I ′ ,Φ(1 , = 1, the set α − ((0 , ∞ ) ) ⊆ T × I ′ , and Φ does not attain thevalue 1 on α − ((0 , ∞ ) ) . Let U = V ∗ be the coisometry from Theorem 2.1 associated to Φand defined in (2.6). Then(6.1) Φ( z ) = A + B (1 − E z D ) − E z C for z ∈ D and by Theorem 3.2, (1 − E τ D ) − exists for all τ ∈ T × I ′ .This implies that (6.1) extends to all τ ∈ T × I ′ , including (1 , , = 1 and (1 − D ) − exists, standard information about inversesfor block 2 × − U is invertible.Since U is a co-isometry and 1 − U is invertible, the von Neumann-Wolddecomposition implies that U is unitary.Fix any w ∈ Π ∪ (0 , ∞ ) , so that w = α ( z ) for some z ∈ D ∪ α − ((0 , ∞ ) ). Then we can apply Theorems 4.1 and 4.2 with Z = E z and W = E w to conclude that(6.2) F ( w ) = T − T ( T + E w ) − T , where T = i (1 + U )(1 − U ) − . Since U is unitary, T is self-adjoint andsince ( T + E w ) − exists for w ∈ (0 , ∞ ) , T must be positive semi-definite. Observe that (6.2) has a natural extension to a map sendingall pairs of matrix inputs with positive imaginary part to outputs withpositive imaginary part, see for example [50, Theorem 5.7]. Since T is positive semidefinite, (6.2) extends further to all pairs of positivematrices as inputs for w , w . As the cone of pairs of positive matricesis a free, convex set, the noncommutative L¨owner theorem, see [50,
ONCRETE REALIZATIONS AND THE MCC 35
Theorem 1.7] as well as [46, 52, 49], implies that F is globally matrixmonotone on (0 , ∞ ) . (cid:3) Perspective functions.
Define a commutative perspectivefunction f to be a locally matrix monotone function on an open cone C ⊆ (0 , ∞ ) d such that f ( tz ) = tf ( z ) when t ∈ R + . Perspective func-tions appear in the work of Andˆo and Kubo in the context of monotonefunctions and operator means via L¨owner’s theorem, and in the convexoptimization regime in a series of papers by Effros, Hansen, and others.In particular, Effros and Hansen prove that convex non-commutativeperspectives arise from convex commutative perspectives. See, e.g.[37, 25, 24, 26].
Theorem 6.2. If f is a commutative perspective function on an opencone C ⊆ (0 , ∞ ) d , then f is globally matrix monotone on (0 , ∞ ) d .Proof. By the Theorem 1.1 (the commutative L¨owner theorem), f hasan analytic continuation as a Pick-Agler function f on the poly upperhalf plane Π d . Since C ⊆ R d is open, the identity theorem impliesthat this analytic continuation is unique on Π d . For any t ∈ R + , con-sider the Pick-Agler function g ( z ) = tf ( z/t ). Because f is positivelyhomogenous on C , f = g on C and by the uniqueness of the exten-sion, f = g on Π d . Thus, f is positively homogeneous on Π d , whichimmediately implies that the non-tangential value of f at 0 is 0.Now we show that f has a useful Nevanlinna representation. To doso, we need to show that f is sufficiently well behaved at 0 (that is, f has a carapoint at 0 in the language of [6]). Let H ( z ) = f ( − /z ).Then lim inf y →∞ y | H ( iy, . . . , iy ) | = lim inf y →∞ y (cid:12)(cid:12)(cid:12) f ( i y , . . . , i y ) (cid:12)(cid:12)(cid:12) = lim inf y →∞ | f ( i, . . . , i ) | < ∞ . Given this, Theorem 1.6 in [6] says that there must exist a Hilbert space H , a densely-defined self-adjoint operator A on H , positive semidefinitecontractions Y , . . . , Y d summing to 1 on H , and a vector ν ∈ H so thatfor all z ∈ Π d , H ( z ) = D ( A − X z i Y i ) − ν, ν E H . Therefore, the same objects give a representation of f by f ( z ) = D ( A + X z − i Y i ) − ν, ν E H for all z ∈ Π d . So, since f ( z ) = tf ( z/t ), f ( z ) = h ( A + X z − i Y i ) − ν, ν i H = h ( t A + X z − i Y i ) − ν, ν i H . By letting t → ∞ , we can assume A = 0. Then since ( P z − i Y i ) − iswell defined for all d tuples of positive matrices, the noncommutativeL¨owner theorem [50] implies that f is globally matrix monotone on(0 , ∞ ) d . (cid:3) References [1] J. Agler. Some interpolation theorems of Nevanlinna-Pick type. Preprint, 1988.[2] J. Agler. On the representation of certain holomorphic functions defined ona polydisc. In
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