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The von Neumann inequality for 3 × 3 matrices

The von Neumann inequality for 3 × 3 matrices Abstract This note details how recent work of Kosiński on the three-point Pick interpolation problem on the polydisk can be used to prove the von Neumann inequality for $$d$$-tuples of commuting $$3\times 3$$ contractive matrices. 1. Introduction The purpose of this note is to explain how recent results of Kosiński [14] provide a proof of the von Neumann inequality for $$d$$-tuples of $$3\times 3$$ commuting contractive matrices. Definition 1.1 A $$d$$-tuple of pairwise commuting contractive matrices or operators $$T = (T_1,\ldots , T_d)$$ satisfies the von Neumann inequality if, for every $$p \in \mathbb {C}[z_1,\ldots ,z_d]$$,   \[ \|p(T) \| \leqslant \sup_{z \in \mathbb{T}^d} |p(z)|. \] Recall that contractive means the Hilbert space operator norm $$\|T_j\| \leqslant 1$$ and $$\mathbb {T}^d$$ is the unit $$d$$-torus in $$\mathbb {C}^d$$. The von Neumann inequality holds for a single contractive operator, von Neumann's original result [16], as well as for a pair of commuting contractions, a result of Andô [2]. For $$d>2,$$ there are known examples of $$d$$-tuples of commuting contractions for which the von Neumann inequality fails. Varopoulos [18] proved the existence of counterexamples with a probabilistic argument and later Kaijser and Varopoulos (see addendum to [18]) as well as Crabb and Davie [4] found explicit counterexamples. These counterexamples were all given by finite matrices. It turns out that the von Neumann inequality holds for $$d$$-tuples of $$2\times 2$$ commuting contractive matrices; this result is essentially equivalent to the Schwarz lemma on the polydisk. On the other hand, Holbrook [11] has found a 3-tuple of $$4\times 4$$ matrices that fail the von Neumann inequality. A great deal of effort has been expended to answer the following question: Does the von Neumann inequality hold for $$d$$-tuples of $$3\times 3$$ commuting contractive matrices? For evidence of interest in this question see [3, 11, 15]. See also the thesis [8], which addressed some special cases of this question. Recent work of Kosiński proves that the answer is yes. Theorem 1.2 The von Neumann inequality holds for$$d$$-tuples of$$3\times 3$$commuting contractive matrices. This is also interesting in light of the fact that there exists a 4-tuple of $$3\times 3$$ commuting contractions that does not dilate to a commuting 4-tuple of unitary operators [3]. Possessing a unitary dilation is a much stronger property than satisfying a von Neumann inequality; in fact, it means that a von Neumann inequality holds for matrix-valued polynomials of all matrix sizes. Indeed, the 4-tuple of $$3\times 3$$ matrices of [3] fails the von Neumann inequality for $$2\times 1$$ matrix-valued polynomials. This leaves the following question unresolved: Does every 3-tuple $$T$$ of $$3\times 3$$ contractive matrices have a unitary dilation? Equivalently, for every such $$T$$ and every matrix-valued polynomial $$P=(p_{j,k}) \in \mathbb {C}^{M\times N}[z_1,z_2,z_3]$$, do we have   \[ \|(p_{j,k}(T))_{j,k}\| \leqslant \sup\{\|P(z)\|: z \in \mathbb{D}^3\}? \] If not, what are the minimal matrix sizes $$M\times N$$ for which this does hold? For some context, we point out that many of the von Neumann inequalities cited above have dilation counterparts: the Sz.-Nagy dilation theorem vis-à-vis von Neumann's inequality, Andô's dilation theorem (Andô's actual result in [2]), $$d$$-tuples of $$2\times 2$$ commuting contractive matrices always dilate [6, 10]. The situation of a von Neumann inequality without a dilation theorem is not uncommon though, and it is closely related to the distinction between spectral sets and complete spectral sets. See [5] for more information. In the next two sections, we point out some known reductions and then explain how Kosiński's work proves Theorem 1.2. In the third and final section, we explain some other operator theory-related consequences of [14]. Namely, solvable three-point Pick interpolation problems on the polydisk can always be solved with a rational inner function in the Schur–Agler class. 2. Reductions Let $$T=(T_1,\ldots , T_d)$$ be a $$d$$-tuple of $$3\times 3$$ commuting contractive matrices. To start with, we can assume that the matrices are strict contractions ($$\|T_j\| < 1$$, $$j=1,\ldots , d$$) and then the theorem will follow by continuity. Next, a $$d$$-tuple of $$3\times 3$$ commuting matrices can always be perturbed to a simultaneously diagonalizable $$d$$-tuple of commuting matrices. The reference [12] points out several places where this is proven; see [7, 9, 15]. After adjusting our operators to be nicer, we will replace polynomials with functions that are less nice. If $$T$$ is now a $$d$$-tuple of commuting simultaneously diagonalizable strictly contractive $$3\times 3$$ matrices, then it suffices to prove   \[ \|f(T)\| \leqslant 1 \] for all $$f:\mathbb {D}^d \to \mathbb {D}$$ holomorphic on the unit $$d$$-dimensional polydisk. This follows from the fact that such holomorphic functions can be approximated locally uniformly on $$\mathbb {D}^d$$ by polynomials $$p \in \mathbb {C}[z_1,\ldots , z_d]$$ with supremum norm at most 1 on $$\mathbb {D}^d$$ (or $$\mathbb {T}^d$$ by the maximum principle). See Rudin [17, p. 126]. Using bounded holomorphic functions makes it possible to apply Möbius transformations to the matrices $$T_1,\ldots , T_d$$ in order to force one of the joint eigenvalues of $$T$$ to be $$0 \in \mathbb {C}^d$$, while still maintaining all other properties of $$T$$. We can also apply a Möbius transformation to $$f:\mathbb {D}^d \to \mathbb {D}$$ and assume $$f(0)=0$$. Let $$0,z,w \in \mathbb {D}^d$$ be the joint eigenvalues of $$T$$ with corresponding eigenvectors $$e, u, v\in \mathbb {C}^3$$. Then, $$f(T)$$ is the $$3\times 3$$ matrix with eigenvalues $$0, \sigma =f(z), \tau =f(w) \in \mathbb {D}$$ and eigenvectors $$e,u,v$$. It now becomes of interest to understand all holomorphic functions $$g:\mathbb {D}^d \to \mathbb {D}$$ which solve the following interpolation problem:   \begin{equation} \begin{aligned} 0 &\longmapsto 0, \\ z &\longmapsto \sigma, \\ w &\longmapsto \tau. \end{aligned} \end{equation} (2.1) Theorem 1.2 will follow from the next result, which is explained in the next section. Theorem 2.1 (Kosiński) If the interpolation problem (2.1) can be solved with$$g:\mathbb {D}^d\to \mathbb {D}$$holomorphic, then there exist holomorphic$$F_1,F_2:\mathbb {D}^2 \to \mathbb {D}$$such that (2.1) can be solved with a function of the form  \begin{equation} F(z) = F_1(F_2(z_1,z_2),z_3) \end{equation} (2.2)after possibly permuting the variables. Indeed, by Andô's inequality $$S = F_2(T_1,T_2)$$ is a contraction commuting with $$T_3$$ and therefore $$F(T) = F_1(S,T_3)$$ is a contraction equal to $$f(T),$$ as above. This proves Theorem 1.2 given Theorem 2.1. 3. The three-point Pick problem on the polydisk Theorem 2.1 follows from work in [14] after making some further reductions to put us in the most interesting situation (that of extremal and non-degenerate interpolation problems). First, it is worth pointing out that the two-point Pick problem on the polydisk is simple to analyze using one-dimensional slices and the Schwarz lemma. It is possible to solve   \begin{align*} 0 \in \mathbb{D}^d &\longmapsto 0 \in \mathbb{D}, \\ z \in \mathbb{D}^d &\longmapsto \sigma \in \mathbb{D} \end{align*} with an analytic $$f:\mathbb {D}^d \to \overline {\mathbb {D}}$$ if and only if $$|\sigma | \leqslant \max _{j} |z_j|$$. However, even simple problems, such as $$(0,0) \mapsto 0$$ and $$(\frac {1}{2}, \frac {1}{2}) \mapsto \frac {1}{2}$$, will have many interpolants $$f$$ due to the geometry of the polydisk. See Section 11.6 of the book [1]. For these reasons, it is useful to perturb the nodes $$0, z=(z_1,\ldots , z_d), w=(w_1,\ldots ,w_d)$$ into a more generic position. Let $$\rho (a,b) = |{(a-b)}/{(1-\bar {a}b)}|$$ be the pseudo-hyperbolic distance on the unit disk. Perturb $$z,w$$ so that all of the quantities below are distinct, yet $$T$$ is still strictly contractive:   \[ |z_1|,\ldots, |z_d|, |w_1|,\ldots, |w_d|, \rho(z_1,w_1),\ldots, \rho(z_d,w_d). \] Functions as in (2.2) form a normal family, so we can approximate the less generic interpolation problems by the generic ones. The interpolation problem 2.1 is said to be extremal if it cannot be solved with a holomorphic function $$g$$ satisfying $$\sup _{\mathbb {D}^d} |g| <1$$. There is no harm in multiplying $$\sigma , \tau $$ by $$r>1$$ if necessary to force the problem to be extremal. The interpolation problem 2.1 is said to be non-degenerate if no two-point subproblem is extremal. If a two-point subproblem is extremal (the degenerate case), then one of the following holds:   \[ |\sigma| = \max_{j=1,\ldots, d} |z_j| \quad \hbox{or} \quad |\tau| = \max_{j=1,\ldots, d} |w_j| \quad \hbox{or} \quad \rho(\sigma, \tau) = \max_{j=1,\ldots, d} \rho(z_j,w_j). \] By our genericity assumption, whichever maximum occurs above will occur at a unique $$j$$ and this forces the solution function to be unique and to depend on one variable. We provide some details in Lemma 3.1 at the end of this section. Thus, in the degenerate case we can certainly solve with a function of the form (2.2) and we may now assume that our interpolation problem is non-degenerate. To finish, we may quote appropriate results from [14]. Lemma 3 of [14] states that, for $$d=3$$, if the interpolation problem (2.1) is extremal, non-degenerate and strictly three-dimensional, then it can be solved with a function of the form (2.2). Strictly three-dimensional means the problem can be solved with a function depending on three variables but not with a function depending only on two variables. We may certainly assume that we are in the strictly three-dimensional case since otherwise there is nothing to prove. For $$d>3$$, Lemma 5 of [14] states that if (2.1) is extremal and non-degenerate, then, after permuting variables if necessary, the problem can be solved with a function of the form (2.2). This completes our explanation of Kosiński's Theorem 2.1. We used the following fact above. Lemma 3.1 Suppose$$f:\mathbb {D}^d \to \mathbb {D}$$is holomorphic,$$f(0)=0$$and there exists$$w \in \mathbb {D}^d$$such that$$f(w) = w_1$$and$$|w_1| > |w_j|$$for all$$j\ne 1$$. Then,$$f(z) \equiv z_1$$for all$$z \in \mathbb {D}^d$$. Proof For $$\zeta \in \mathbb {D}$$, let $$h(\zeta ) = f(({\zeta }/{w_1})w)$$. Then, $$h(0)=0, h(w_1) = w_1$$ and therefore, by the classical Schwarz lemma, $$h(\zeta ) \equiv \zeta $$. This implies   \[ \sum_{j=1}^{d} \frac{\partial f}{\partial z_j}(0) \frac{w_j}{w_1} = 1. \] By [17, p. 179], $$\sum _{j=1}^{d} |({\partial f}/{\partial z_j})(0)| \leqslant 1$$. Since $$|w_1| > |w_j|$$ for $$j\ne 1$$, this can happen only if $$({\partial f}/{\partial z_1})(0) = 1$$. This implies $$f(\zeta ,0,\ldots ,0) \equiv \zeta $$. By Lemma 3.2 of [13], this implies $$f(z) \equiv z_1$$. 4. The Schur–Agler class The Schur–Agler class on the polydisk $$\mathbb {D}^d$$ consists of holomorphic functions $$f:\mathbb {D}^d \to \overline {\mathbb {D}}$$ such that   \[ \|f(T)\| \leqslant 1 \] for all $$d$$-tuples $$T$$ of commuting strictly contractive operators. Functions of the form (2.2) are certainly in the Schur–Agler class; indeed, the argument after the statement of Theorem 2.1 proves this. Thus, three-point Pick interpolation problems on the polydisk can be solved with functions in the Schur–Agler class. However, the Schur–Agler class has a well-known interpolation theorem due to Agler; see [1, Theorems 11.49 and 11.90]. Combining these observations, we get the following. Theorem 4.1 Given$$z_1,z_2,z_3 \in \mathbb {D}^d$$and$$t_1,t_2,t_3 \in \mathbb {D},$$there exists a holomorphic function$$f:\mathbb {D}^d\to \mathbb {D}$$satisfying$$f(z_j) = t_j$$for$$j=1,2,3$$if and only if there exist positive semi-definite$$3\times 3$$matrices$$\Gamma ^1, \Gamma ^2,\ldots , \Gamma ^d$$such that  \[ 1-t_j \overline{t_k} = \sum_{n=1}^{d} (1-z_j^n \overline{z_k^n}) \Gamma_{j,k}^n \]for$$j,k=1,2,3$$. Here superscripts are used to denote components of$$z_j \in \mathbb {D}^d$$. This result can be used to prove that every solvable three-point Pick problem can be solved with a rational inner function in the Schur–Agler class. The paper [14] already proves this in the case of non-degenerate extremal problems but then the above general machinery can be used to show that every solvable problem, whether extremal or not, has a rational inner solution. References 1 Agler J. McCarthy J. E., Pick interpolation and Hilbert function spaces , Graduate Studies in Mathematics 44 ( American Mathematical Society, Providence, RI, 2002). 2 Andô T., ‘ On a pair of commutative contractions’, Acta Sci. Math. (Szeged)  24 ( 1963) 88– 90. 3 Choi M.-D. Davidson K. R., ‘ A $$3\times 3$$ dilation counterexample’, Bull. London Math. Soc.  45 ( 2013) 511– 519. Google Scholar CrossRef Search ADS   4 Crabb M. J. Davie A. M., ‘ von Neumann's inequality for Hilbert space operators’, Bull. London Math. Soc.  7 ( 1975) 49– 50. Google Scholar CrossRef Search ADS   5 Dritschel M. A. McCullough S., ‘ The failure of rational dilation on a triply connected domain’, J. Amer. Math. Soc.  18 ( 2005) 873– 918. Google Scholar CrossRef Search ADS   6 Drury S. W., ‘ Remarks on von Neumann's inequality’, Banach spaces, harmonic analysis, and probability theory, Storrs, CT, 1980/1981 , Lecture Notes in Mathematics 995 ( Springer, Berlin, 1983) 14– 32. 7 Gerstenhaber M., ‘ On dominance and varieties of commuting matrices’, Ann. of Math. (2)  73 ( 1961) 324– 348. Google Scholar CrossRef Search ADS   8 Gupta R., The Carathéodory–Fejér interpolation problems and the von Neumann inequality  ( Indian Institute of Science, Bangalore, 2015). 9 Guralnick R. M., ‘ A note on commuting pairs of matrices’, Linear Multilinear Algebra  31 ( 1992) 71– 75. Google Scholar CrossRef Search ADS   10 Holbrook J. A., ‘ Inequalities of von Neumann type for small matrices’, Function spaces, Edwardsville, IL, 1990 , Lecture Notes in Pure and Applied Mathematics 136 ( Dekker, New York, 1992) 189– 193. 11 Holbrook J. A., ‘ Schur norms and the multivariate von Neumann inequality’, Recent advances in operator theory and related topics, Szeged, 1999 , Operator Theory: Advances and Applications 127 ( Birkhäuser, Basel, 2001) 375– 386. 12 Holbrook J. Omladič M., ‘ Approximating commuting operators’, Linear Algebra Appl.  327 ( 2001) 131– 149. Google Scholar CrossRef Search ADS   13 Knese G., ‘ A refined Agler decomposition and geometric applications’, Indiana Univ. Math. J.  60 ( 2011) 1831– 1841. Google Scholar CrossRef Search ADS   14 Kosiński Ł., ‘ Three-point Nevanlinna Pick problem in the polydisc’, Proc. London Math. Soc.  111 (2015) 887–910. 15 Lotto B. A., ‘ von Neumann's inequality for commuting, diagonalizable contractions. I’, Proc. Amer. Math. Soc.  120 ( 1994) 889– 895. 16 von Neumann J., ‘ Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes’, Math. Nachr.  4 ( 1951) 258– 281 ( German). Google Scholar CrossRef Search ADS   17 Rudin W., Function theory in polydiscs  ( W. A. Benjamin, New York–Amsterdam, 1969). 18 Varopoulos N. Th., ‘ On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory’, J. Funct. Anal.  16 ( 1974) 83– 100. Google Scholar CrossRef Search ADS   © 2015 London Mathematical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Oxford University Press

The von Neumann inequality for 3 × 3 matrices

Bulletin of the London Mathematical Society , Volume 48 (1) – Nov 30, 2015

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Publisher
Oxford University Press
Copyright
© 2015 London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/bdv087
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Abstract

Abstract This note details how recent work of Kosiński on the three-point Pick interpolation problem on the polydisk can be used to prove the von Neumann inequality for $$d$$-tuples of commuting $$3\times 3$$ contractive matrices. 1. Introduction The purpose of this note is to explain how recent results of Kosiński [14] provide a proof of the von Neumann inequality for $$d$$-tuples of $$3\times 3$$ commuting contractive matrices. Definition 1.1 A $$d$$-tuple of pairwise commuting contractive matrices or operators $$T = (T_1,\ldots , T_d)$$ satisfies the von Neumann inequality if, for every $$p \in \mathbb {C}[z_1,\ldots ,z_d]$$,   \[ \|p(T) \| \leqslant \sup_{z \in \mathbb{T}^d} |p(z)|. \] Recall that contractive means the Hilbert space operator norm $$\|T_j\| \leqslant 1$$ and $$\mathbb {T}^d$$ is the unit $$d$$-torus in $$\mathbb {C}^d$$. The von Neumann inequality holds for a single contractive operator, von Neumann's original result [16], as well as for a pair of commuting contractions, a result of Andô [2]. For $$d>2,$$ there are known examples of $$d$$-tuples of commuting contractions for which the von Neumann inequality fails. Varopoulos [18] proved the existence of counterexamples with a probabilistic argument and later Kaijser and Varopoulos (see addendum to [18]) as well as Crabb and Davie [4] found explicit counterexamples. These counterexamples were all given by finite matrices. It turns out that the von Neumann inequality holds for $$d$$-tuples of $$2\times 2$$ commuting contractive matrices; this result is essentially equivalent to the Schwarz lemma on the polydisk. On the other hand, Holbrook [11] has found a 3-tuple of $$4\times 4$$ matrices that fail the von Neumann inequality. A great deal of effort has been expended to answer the following question: Does the von Neumann inequality hold for $$d$$-tuples of $$3\times 3$$ commuting contractive matrices? For evidence of interest in this question see [3, 11, 15]. See also the thesis [8], which addressed some special cases of this question. Recent work of Kosiński proves that the answer is yes. Theorem 1.2 The von Neumann inequality holds for$$d$$-tuples of$$3\times 3$$commuting contractive matrices. This is also interesting in light of the fact that there exists a 4-tuple of $$3\times 3$$ commuting contractions that does not dilate to a commuting 4-tuple of unitary operators [3]. Possessing a unitary dilation is a much stronger property than satisfying a von Neumann inequality; in fact, it means that a von Neumann inequality holds for matrix-valued polynomials of all matrix sizes. Indeed, the 4-tuple of $$3\times 3$$ matrices of [3] fails the von Neumann inequality for $$2\times 1$$ matrix-valued polynomials. This leaves the following question unresolved: Does every 3-tuple $$T$$ of $$3\times 3$$ contractive matrices have a unitary dilation? Equivalently, for every such $$T$$ and every matrix-valued polynomial $$P=(p_{j,k}) \in \mathbb {C}^{M\times N}[z_1,z_2,z_3]$$, do we have   \[ \|(p_{j,k}(T))_{j,k}\| \leqslant \sup\{\|P(z)\|: z \in \mathbb{D}^3\}? \] If not, what are the minimal matrix sizes $$M\times N$$ for which this does hold? For some context, we point out that many of the von Neumann inequalities cited above have dilation counterparts: the Sz.-Nagy dilation theorem vis-à-vis von Neumann's inequality, Andô's dilation theorem (Andô's actual result in [2]), $$d$$-tuples of $$2\times 2$$ commuting contractive matrices always dilate [6, 10]. The situation of a von Neumann inequality without a dilation theorem is not uncommon though, and it is closely related to the distinction between spectral sets and complete spectral sets. See [5] for more information. In the next two sections, we point out some known reductions and then explain how Kosiński's work proves Theorem 1.2. In the third and final section, we explain some other operator theory-related consequences of [14]. Namely, solvable three-point Pick interpolation problems on the polydisk can always be solved with a rational inner function in the Schur–Agler class. 2. Reductions Let $$T=(T_1,\ldots , T_d)$$ be a $$d$$-tuple of $$3\times 3$$ commuting contractive matrices. To start with, we can assume that the matrices are strict contractions ($$\|T_j\| < 1$$, $$j=1,\ldots , d$$) and then the theorem will follow by continuity. Next, a $$d$$-tuple of $$3\times 3$$ commuting matrices can always be perturbed to a simultaneously diagonalizable $$d$$-tuple of commuting matrices. The reference [12] points out several places where this is proven; see [7, 9, 15]. After adjusting our operators to be nicer, we will replace polynomials with functions that are less nice. If $$T$$ is now a $$d$$-tuple of commuting simultaneously diagonalizable strictly contractive $$3\times 3$$ matrices, then it suffices to prove   \[ \|f(T)\| \leqslant 1 \] for all $$f:\mathbb {D}^d \to \mathbb {D}$$ holomorphic on the unit $$d$$-dimensional polydisk. This follows from the fact that such holomorphic functions can be approximated locally uniformly on $$\mathbb {D}^d$$ by polynomials $$p \in \mathbb {C}[z_1,\ldots , z_d]$$ with supremum norm at most 1 on $$\mathbb {D}^d$$ (or $$\mathbb {T}^d$$ by the maximum principle). See Rudin [17, p. 126]. Using bounded holomorphic functions makes it possible to apply Möbius transformations to the matrices $$T_1,\ldots , T_d$$ in order to force one of the joint eigenvalues of $$T$$ to be $$0 \in \mathbb {C}^d$$, while still maintaining all other properties of $$T$$. We can also apply a Möbius transformation to $$f:\mathbb {D}^d \to \mathbb {D}$$ and assume $$f(0)=0$$. Let $$0,z,w \in \mathbb {D}^d$$ be the joint eigenvalues of $$T$$ with corresponding eigenvectors $$e, u, v\in \mathbb {C}^3$$. Then, $$f(T)$$ is the $$3\times 3$$ matrix with eigenvalues $$0, \sigma =f(z), \tau =f(w) \in \mathbb {D}$$ and eigenvectors $$e,u,v$$. It now becomes of interest to understand all holomorphic functions $$g:\mathbb {D}^d \to \mathbb {D}$$ which solve the following interpolation problem:   \begin{equation} \begin{aligned} 0 &\longmapsto 0, \\ z &\longmapsto \sigma, \\ w &\longmapsto \tau. \end{aligned} \end{equation} (2.1) Theorem 1.2 will follow from the next result, which is explained in the next section. Theorem 2.1 (Kosiński) If the interpolation problem (2.1) can be solved with$$g:\mathbb {D}^d\to \mathbb {D}$$holomorphic, then there exist holomorphic$$F_1,F_2:\mathbb {D}^2 \to \mathbb {D}$$such that (2.1) can be solved with a function of the form  \begin{equation} F(z) = F_1(F_2(z_1,z_2),z_3) \end{equation} (2.2)after possibly permuting the variables. Indeed, by Andô's inequality $$S = F_2(T_1,T_2)$$ is a contraction commuting with $$T_3$$ and therefore $$F(T) = F_1(S,T_3)$$ is a contraction equal to $$f(T),$$ as above. This proves Theorem 1.2 given Theorem 2.1. 3. The three-point Pick problem on the polydisk Theorem 2.1 follows from work in [14] after making some further reductions to put us in the most interesting situation (that of extremal and non-degenerate interpolation problems). First, it is worth pointing out that the two-point Pick problem on the polydisk is simple to analyze using one-dimensional slices and the Schwarz lemma. It is possible to solve   \begin{align*} 0 \in \mathbb{D}^d &\longmapsto 0 \in \mathbb{D}, \\ z \in \mathbb{D}^d &\longmapsto \sigma \in \mathbb{D} \end{align*} with an analytic $$f:\mathbb {D}^d \to \overline {\mathbb {D}}$$ if and only if $$|\sigma | \leqslant \max _{j} |z_j|$$. However, even simple problems, such as $$(0,0) \mapsto 0$$ and $$(\frac {1}{2}, \frac {1}{2}) \mapsto \frac {1}{2}$$, will have many interpolants $$f$$ due to the geometry of the polydisk. See Section 11.6 of the book [1]. For these reasons, it is useful to perturb the nodes $$0, z=(z_1,\ldots , z_d), w=(w_1,\ldots ,w_d)$$ into a more generic position. Let $$\rho (a,b) = |{(a-b)}/{(1-\bar {a}b)}|$$ be the pseudo-hyperbolic distance on the unit disk. Perturb $$z,w$$ so that all of the quantities below are distinct, yet $$T$$ is still strictly contractive:   \[ |z_1|,\ldots, |z_d|, |w_1|,\ldots, |w_d|, \rho(z_1,w_1),\ldots, \rho(z_d,w_d). \] Functions as in (2.2) form a normal family, so we can approximate the less generic interpolation problems by the generic ones. The interpolation problem 2.1 is said to be extremal if it cannot be solved with a holomorphic function $$g$$ satisfying $$\sup _{\mathbb {D}^d} |g| <1$$. There is no harm in multiplying $$\sigma , \tau $$ by $$r>1$$ if necessary to force the problem to be extremal. The interpolation problem 2.1 is said to be non-degenerate if no two-point subproblem is extremal. If a two-point subproblem is extremal (the degenerate case), then one of the following holds:   \[ |\sigma| = \max_{j=1,\ldots, d} |z_j| \quad \hbox{or} \quad |\tau| = \max_{j=1,\ldots, d} |w_j| \quad \hbox{or} \quad \rho(\sigma, \tau) = \max_{j=1,\ldots, d} \rho(z_j,w_j). \] By our genericity assumption, whichever maximum occurs above will occur at a unique $$j$$ and this forces the solution function to be unique and to depend on one variable. We provide some details in Lemma 3.1 at the end of this section. Thus, in the degenerate case we can certainly solve with a function of the form (2.2) and we may now assume that our interpolation problem is non-degenerate. To finish, we may quote appropriate results from [14]. Lemma 3 of [14] states that, for $$d=3$$, if the interpolation problem (2.1) is extremal, non-degenerate and strictly three-dimensional, then it can be solved with a function of the form (2.2). Strictly three-dimensional means the problem can be solved with a function depending on three variables but not with a function depending only on two variables. We may certainly assume that we are in the strictly three-dimensional case since otherwise there is nothing to prove. For $$d>3$$, Lemma 5 of [14] states that if (2.1) is extremal and non-degenerate, then, after permuting variables if necessary, the problem can be solved with a function of the form (2.2). This completes our explanation of Kosiński's Theorem 2.1. We used the following fact above. Lemma 3.1 Suppose$$f:\mathbb {D}^d \to \mathbb {D}$$is holomorphic,$$f(0)=0$$and there exists$$w \in \mathbb {D}^d$$such that$$f(w) = w_1$$and$$|w_1| > |w_j|$$for all$$j\ne 1$$. Then,$$f(z) \equiv z_1$$for all$$z \in \mathbb {D}^d$$. Proof For $$\zeta \in \mathbb {D}$$, let $$h(\zeta ) = f(({\zeta }/{w_1})w)$$. Then, $$h(0)=0, h(w_1) = w_1$$ and therefore, by the classical Schwarz lemma, $$h(\zeta ) \equiv \zeta $$. This implies   \[ \sum_{j=1}^{d} \frac{\partial f}{\partial z_j}(0) \frac{w_j}{w_1} = 1. \] By [17, p. 179], $$\sum _{j=1}^{d} |({\partial f}/{\partial z_j})(0)| \leqslant 1$$. Since $$|w_1| > |w_j|$$ for $$j\ne 1$$, this can happen only if $$({\partial f}/{\partial z_1})(0) = 1$$. This implies $$f(\zeta ,0,\ldots ,0) \equiv \zeta $$. By Lemma 3.2 of [13], this implies $$f(z) \equiv z_1$$. 4. The Schur–Agler class The Schur–Agler class on the polydisk $$\mathbb {D}^d$$ consists of holomorphic functions $$f:\mathbb {D}^d \to \overline {\mathbb {D}}$$ such that   \[ \|f(T)\| \leqslant 1 \] for all $$d$$-tuples $$T$$ of commuting strictly contractive operators. Functions of the form (2.2) are certainly in the Schur–Agler class; indeed, the argument after the statement of Theorem 2.1 proves this. Thus, three-point Pick interpolation problems on the polydisk can be solved with functions in the Schur–Agler class. However, the Schur–Agler class has a well-known interpolation theorem due to Agler; see [1, Theorems 11.49 and 11.90]. Combining these observations, we get the following. Theorem 4.1 Given$$z_1,z_2,z_3 \in \mathbb {D}^d$$and$$t_1,t_2,t_3 \in \mathbb {D},$$there exists a holomorphic function$$f:\mathbb {D}^d\to \mathbb {D}$$satisfying$$f(z_j) = t_j$$for$$j=1,2,3$$if and only if there exist positive semi-definite$$3\times 3$$matrices$$\Gamma ^1, \Gamma ^2,\ldots , \Gamma ^d$$such that  \[ 1-t_j \overline{t_k} = \sum_{n=1}^{d} (1-z_j^n \overline{z_k^n}) \Gamma_{j,k}^n \]for$$j,k=1,2,3$$. Here superscripts are used to denote components of$$z_j \in \mathbb {D}^d$$. This result can be used to prove that every solvable three-point Pick problem can be solved with a rational inner function in the Schur–Agler class. 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Published: Nov 30, 2015

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