The class of Schur-Agler functions over a domain ${\mathcal D} \subset {\mathbb C}^{d}$ is defined as the class of holomorphic operator-valued functions on ${\mathcal D}$ for which a certain von Neumann inequality is satisfied when a commuting tuple of operators satisfying a certain polynomial norm inequality is plugged in for the variables. Such functions are alternatively characterized as those having a linear-fractional presentation which identifies them as transfer functions of a certain type of conservative structured multidimensional linear system. There now has been introduced a noncommutative version of the Schur-Agler class which consists of formal power series in noncommuting indeterminants satisfying a noncommutative version of the von Neumann inequality when a tuple of operators (not necessarily commuting) coming from a noncommutative operator ball are plugged in for the formal indeterminants. Formal power series in this noncommutative Schur-Agler class in turn are characterized as those having a certain linear-fractional presentation in noncommuting variables identifying them as transfer functions of a recently introduced class of conservative structure multidimensional linear systems having evolution along a free semigroup rather than along an integer lattice. The purpose of this paper is to extend the previously developed interpolation theory for the commutative Schur-Agler class to this noncommutative setting.
The Schur-Agler class consists of functions over a domain satisfying an appropriate von Neumann inequality. Originally defined over the polydisk, the idea has been extended to general domains in multivariable complex Euclidean space with matrix polynomial defining function as well as to certain multivariable noncommutative-operator domains with a noncommutative linear-pencil defining function. Still more recently there has emerged a free noncommutative function theory (functions of noncommuting matrix variables respecting direct sums and similaritytransformations). The purpose of the present p…
Given a domain $Ω$ in $\mathbb{C}^m$, and a finite set of points $z_1,\ldots, z_n\in Ω$ and $w_1,\ldots, w_n\in \mathbb{D}$ (the open unit disc in the complex plane), the $Pick\, interpolation\, problem$ asks when there is a holomorphic function $f:Ω\rightarrow \overline{\mathbb{D}}$ such that $f(z_i)=w_i,1\leq i\leq n$. Pick gave a condition on the data $\{z_i, w_i:1\leq i\leq n\}$ for such an $interpolant$ to exist if $Ω=\mathbb{D}$. Nevanlinna characterized all possible functions $f$ that $interpolate$ the data. We generalize Nevanlinna's result to an arbitrary set $Ω$. In this case, the fu…
We discuss transfer-function realization for multivariable holomorphic functions mapping the unit polydisk or the right polyhalfplane into the operator analogue of either the unit disk or the right halfplane (Schur/Herglotz functions over either the unit polydisk or the right polyhalfplane) which satisfy the appropriate stronger contractive/positive real part condition for the values of these functions on commutative tuples of strict contractions/strictly accretive operators (Schur--Agler/Herglotz--Agler functions over either the unit polydisk or the right polyhalfplane). As originally shown b…
We prove two results with regard to rational inner functions in the Schur-Agler class of the tridisk. Every rational inner function of degree (n,1,1) is in the Schur-Agler class, and every rational inner function of degree (n,m,1) is in the Schur-Agler class after multiplication by a monomial of sufficiently high degree.
The notion of a unitary realization is used to estimate derivatives of arbitrary order of functions in the Schur-Agler class on the polydisk and unit ball.
We call a multivariable polynomial an Agler denominator if it is the denominator of a rational inner function in the Schur-Agler class, an important subclass of the bounded analytic functions on the polydisk. We give a necessary and sufficient condition for a multi-affine, symmetric, and stable polynomial to be an Agler denominator and prove several consequences. We also sharpen a result due to Kummert related to three variable, multi-affine, stable polynomials.
We adapt Sarason's proof of the Julia-Caratheodory theorem to the class of Schur-Agler mappings of the unit ball, obtaining a strengthened form of this theorem. In particular those quantities which appear in the classical theorem and depend only on the component of the mapping in the complex normal direction have K-limits (not just restricted K-limits) at the boundary.
Every two variable rational inner function on the bidisk has a special representation called a transfer function realization. It is well known and related to important ideas in operator theory that this does not extend to three or more variables on the polydisk. We study the class of rational inner functions on the polydisk which do possess a transfer function realization (the Schur-Agler class) and investigate minimality in their representations. Schur-Agler class rational inner functions in three or more variables cannot be represented in a way that is as minimal as two variables might sugge…
A Schur-class function in $d$ variables is defined to be an analytic contractive-operator valued function on the unit polydisk. Such a function is said to be in the Schur--Agler class if it is contractive when evaluated on any commutative $d$-tuple of strict contractions on a Hilbert space. It is known that the Schur--Agler class is a strictly proper subclass of the Schur class if the number of variables $d$ is more than two. The Schur--Agler class is also characterized as those functions arising as the transfer function of a certain type (Givone--Roesser) of conservative multidimensional line…
Given a collection of test functions, one defines the associated Schur-Agler class as the intersection of the contractive multipliers over the collection of all positive kernels for which each test function is a contractive multiplier. We indicate extensions of this framework to the case where the test functions, kernel functions, and Schur-Agler-class functions are allowed to be matrix- or operator-valued. We illustrate the general theory with two examples: (1) the matrix-valued Schur class over a finitely-connected planar domain and (2) the matrix-valued version of the constrained Hardy alge…
In this paper, we prove a noncommutative (nc for short) analog of Schwarz lemma for the nc Schur-Agler class and prove that the regular nc Schur-Agler class and the regular free Herglotz class are homeomorphic. Moreover, we give a characterization of regular free Herglotz functions. As an application, we will show that any regular free Herglotz functions can uniformly be approximated by regular Herglotz free polynomials.
The Bessmertny\uı class consists of rational matrix-valued functions of $d$ complex variables representable as the Schur complement of a block of a linear pencil $A(z)=z_1A_1+\cdots+z_dA_d$ whose coefficients $A_k$ are positive semidefinite matrices. We show that it coincides with the subclass of rational functions in the Herglotz-Agler class over the right poly-halfplane which are homogeneous of degree one and which are Cayley inner. The latter means that such a function is holomorphic on the right poly-halfplane and takes skew-Hermitian matrix values on $(i\mathbb{R})^d$, or equivalently, is…
We identify the set of extreme points and apply Choquet theory to a normalized matrix-measure ball subject to finitely many linear side constraints. As an application we obtain integral representation formulas for the Herglotz class of matrix-valued functions on a finitely-connected planar domain and associated continuous Agler decompositions for the matrix-valued Schur class over the domain. The results give some additional insight into the negative answer to the spectral set problem over such domains recently obtained by Agler-Harland-Raphael and Dritschel-McCullough.
The Schur class, denoted by $\mathcal{S}(\mathbb{D})$, is the set of all functions analytic and bounded by one in modulus in the open unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$, that is \[ \mathcal{S}(\mathbb{D}) = \{φ\in H^\infty(\mathbb{D}): \|φ\|_{\infty} := \sup_{z \in \mathbb{D}} |φ(z)| \leq 1\}. \] The elements of $\mathcal{S}(\mathbb{D})$ are called Schur functions. A classical result going back to I. Schur states: A function $φ: \mathbb{D} \rightarrow \mathbb{C}$ is in $\mathcal{S}(\mathbb{D})$ if and only if there exist a Hilbert space $\mathcal{H}$ and an isometry (know…
We extend Agler's notion of a function algebra defined in terms of test functions to include products, in analogy with the practice in real algebraic geometry, and hence the term preordering in the title. This is done over abstract sets and no additional property, such as analyticity, is assumed. Realization theorems give several equivalent ways of characterizing the unit ball (referred to as the Schur-Agler class) of the function algebras. These typically include, in Agler's terminology, a model (here called an Agler decomposition), a transfer function representation, and an analogue of the v…
A seminal result of Agler proves that the natural de Branges-Rovnyak kernel function associated to a bounded analytic function on the bidisk can be decomposed into two shift-invariant pieces. Agler's decomposition is non-constructive-a problem remedied by work of Ball-Sadosky-Vinnikov, which uses scattering systems to produce Agler decompositions through concrete Hilbert space geometry. This method, while constructive, so far has not revealed the rich structure shown to be present for special classes of functions-inner and rational inner functions. In this paper, we show that most of the impor…
Given a domain $Ω$ in $\mathbb{C}^n$ and a collection of test functions $Ψ$ on $Ω$, we consider the complex-valued $Ψ$-Schur-Agler class associated to the pair $(Ω,\,Ψ)$. In this article, we characterize interpolating sequences for the associated Banach algebra of which the $Ψ$-Schur-Agler class is the closed unit ball. When $Ω$ is the unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$ and the class of test function includes only the identity function on $\mathbb{D}$, the aforementioned algebra is the algebra of bounded holomorphic functions on $\mathbb{D}$ and in this case, our characte…
The Carathéodory problem in the $N$-variable non-commutative Herglotz--Agler class and the Carathéodory--Fejér problem in the $N$-variable non-commutative Schur--Agler class are posed. It is shown that the Carathéodory (resp., Carathéodory--Fejér) problem has a solution if and only if the non-commutative polynomial with given operator coefficients (the data of the problem indexed by an admissible set $Λ$) takes operator values with positive semidefinite real part (resp., contractive operator values) on $N$-tuples of $Λ$-jointly nilpotent contractive $n\times n$ matrices, for all $n\in\mathbb{N…
We generalize our earlier results from \cite{K} on the Bessmertny\uı class of operator-valued functions holomorphic in the open right poly-halfplane which admit representation as a Schur complement of a block of a linear homogeneous operator-valued function with positive semidefinite operator coefficients, to the case of a product of open right matrix halfplanes. Several equivalent characterizations of this generalized Bessmertny\uı class are presented. In particular, its intimate connection with the Agler--Schur class of holomorphic contractive operator-valued functions on the product of matr…
In this paper we solve several problems concerning joint similarity to n-tuples of operators in noncommutative varieties in $[B(\cH)^n]_1$ associated with positive regular free holomorphic functions in $n$ noncommuting variables and with sets of noncommutative polynomials in $n$ indeterminates, where $B(\cH)$ is the algebra of all bounded linear operators on a Hilbert space $\cH$. In particular, if $f=X_1+...+X_n$ and $\cP=\{0\}$, the elements of the corresponding variety can be seen as noncommutative multivariable analogues of Agler's $m$-hypercontractions.
We introduce a noncommutative analogue of the Figá-Talamanca-Herz algebra $A_p(G)$ on the natural predual of the operator space $\frak{M}_{p,cb}$ of completely bounded Schur multipliers on Schatten space $S_p$. We determine the isometric Schur multipliers and prove that the space $\frak{M}_{p}$ of bounded Schur multipliers on Schatten space $S_p$ is the closure in the weak operator topology of the span of isometric multipliers.
A seminal result of Agler characterizes the so-called Schur-Agler class of functions on the polydisk in terms of a unitary colligation transfer function representation. We generalize this to the unit ball of the algebra of multipliers for a family of test functions over a broad class of semigroupoids. There is then an associated interpolation theorem. Besides leading to solutions of the familiar Nevanlinna-Pick and Caratheodory-Fejer interpolation problems and their multivariable commutative and noncommutative generalizations, this approach also covers more exotic examples.
We give a new proof that every linear fractional map of the unit ball induces a bounded composition operator on the standard scale of Hilbert function spaces on the ball, and obtain norm bounds analogous to the standard one-variable estimates. We also show that Cowen's one-variable spectral radius formula extends to these operators. The key observation underlying these results is that every linear fractional map of the ball belongs to the Schur-Agler class.
The class of operator-valued functions which are homogeneous of degree one, holomorphic in the open right polyhalfplane, have positive semidefinite real parts there and take selfadjoint operator values at real points, and its subclass consisting of functions representable in the form of Schur complement of a block of a linear pencil of operators with positive semidefinite operator coefficients, are investigated. The latter subclass is a generalization of the class of characteristic matrix functions of passive 2n-poles considered as functions of impedances of its elements, which was introduced …
Jim Agler revolutionized the area of Pick interpolation with his realization theorem for what is now called the Agler-Schur class for the unit ball in $\mathbb C^d$. We discuss an extension of these results to algebras of functions arising from test functions and the dual notion of a family of reproducing kernels, as well as the related interpolation theorem. When working with test functions, one ideally wants to use as small a collection as possible. Nevertheless, in some situations infinite sets of test functions are unavoidable. When this is the case, certain topological considerations come…
We study representations of inner functions on the bidisc from a fractional linear transformation point of view, and provide sufficient conditions, in terms of colligation matrices, for the existence of two-variable inner functions. Here the sufficient conditions are not necessary in general, and we prove a weak converse for rational inner functions that admit one variable factorization. We present a complete classification of de Branges-Rovnyak kernels on the bidisc (which equally works in the setting of polydisc and the open unit ball of $\mathbb{C}^n$, $n \geq 1$). We also classify, in term…
Schoenberg showed that a function $f:(-1,1)\rightarrow \mathbb{R}$ such that $C=[c_{ij}]_{i,j}$ positive semi-definite implies that $f(C)=[f(c_{ij})]_{i,j}$ is also positive semi-definite must be analytic and have Taylor series coefficients nonnegative at the origin. The Schoenberg theorem is essentially a theorem about the functional calculus arising from the Schur product, the entrywise product of matrices. Two important properties of the Schur product are that the product of two rank one matrices is rank one, and the product of two positive semi-definite matrices is positive semi-definite. …
In this work we investigate semigroups of operators acting on noncommutative $L^p$-spaces. We introduce noncommutative square functions and their connection to sectoriality, variants of Rademacher sectoriality, and $H^\infty$ functional calculus. We discuss several examples of noncommutative diffusion semigroups. This includes Schur multipliers, $q$-Ornstein-Uhlenbeck semigroups, and the noncommutative Poisson semigroup on free groups.
In this paper we begin the study of Schur analysis and de Branges-Rovnyak spaces in the framework of Fueter hyperholomorphic functions. The difference with other approaches is that we consider the class of functions spanned by Appell-like polynomials. This approach is very efficient from various points of view, for example in operator theory, and allows to make connections with the recently developed theory of slice polyanalytic functions. We tackle a number of problems: we describe a Hardy space, Schur multipliers and related results. We also discuss Blaschke functions, Herglotz multipliers a…
Given a function $f: (a,b) \rightarrow \mathbb{R},$ Löwner's theorem states $f$ is monotone when extended to self-adjoint matrices via the functional calculus, if and only if $f$ extends to a self-map of the complex upper half plane. In recent years, several generalizations of Löwner's theorem have been proven in several variables. We use the relaxed Agler, McCarthy and Young theorem on locally matrix monotone functions in several commuting variables to generalize results in the noncommutative case. Specifically, we show that a real free function defined over an operator system must analytical…
We obtain intertwining dilation theorems for noncommutative regular domains D_f and noncommutative varieties V_J of n-tuples of operators, which generalize Sarason and Sz.-Nagy--Foias commutant lifting theorem for commuting contractions. We present several applications including a new proof for the commutant lifting theorem for pure elements in the domain D_f (resp. variety V_J) as well as a Schur type representation for the unit ball of the Hardy algebra associated with the variety V_J. We provide Ando type dilations and inequalities for bi-domains D_f \times D_g and bi-varieties V_J \times V…
In 2010, Lafforgue and de la Salle gave examples of noncommutative Lp-spaces without the operator space approximation property (OAP) and, hence, without the completely bounded approximation property (CBAP). To this purpose, they introduced the property of completely bounded approximation by Schur multipliers on Sp and proved that for p < 4/3 and p > 4 the groups SL(n,Z), with n \geq 3, do not have it. Since for 1 < p < \infty the property of completely bounded approximation by Schur multipliers on Sp is weaker than the approximation property of Haagerup and Kraus (AP), these groups were also t…
Function theory of the hexablock and applications to the tetrablock and Euclidean biball
Mathematics Department, Indian Institute of Technology Bombay（印度理工学院孟买分校数学系）；Statistics and Mathematics unit, Indian Statistical Institute Bangalore（印度统计学院班加罗尔分校统计与数学单元）
We show that the factorization problem $θ(z)=θ_2(z)θ_1(z)$ is solvable in the class of Hilbert space operator-valued functions holomorphic on some neighbourhood of $z=0$ in $\nspace{C}{N}$ and having a zero at $z=0$ (here $θ(z)$ has a multiple zero at $z=0$). Such a factorization problem becomes more complicated if we demand for $θ(z), θ_1(z)$ and $θ_2(z)$ to be Agler--Schur-class functions on the polydisk $\nspace{D}{N}$ and for the factorization identity to hold in $\nspace{D}{N}$. In this case we reduce it to the problem on the existence of a cascade decomposition for certain multiparametri…
In this paper, we initiate the study of sub-pluriharmonic curves in Cuntz-Toeplitz algebras and free pluriharmonic majorants on noncommutative balls. We are lead to a characterization of the noncommutative Hardy space $H^2_{\bf ball}$ in terms of free pluriharmonic majorants, and to a Schur type description of the unit ball of $H^2_{\bf ball}$. These results are used to solve a multivariable commutant lifting problem and provide a description of all solutions.
We introduce a noncommutative analogue of the absolute value of a regular operator acting on a noncommutative $\mathrm{L}^p$-space. We equally prove that two classical operator norms, the regular norm and the decomposable norm are identical. We also describe precisely the regular norm of several classes of regular multipliers. This includes Schur multipliers and Fourier multipliers on some unimodular locally compact groups which can be approximated by discrete groups in various senses. A main ingredient is to show the existence of a bounded projection from the space of completely bounded $\mat…
The theory of Nevanlinna-Pick and Carathéodory-Fejér interpolation for matrix- and operator-valued Schur class functions on the unit disk is now well established. Recent work has produced extensions of the theory to a variety of multivariable settings, including the ball and the polydisk (both commutative and noncommutative versions), as well as a time-varying analogue. Largely independent of this is the recent Nevanlinna-Pick interpolation theorem by P.S. Muhly and B. Solel for an abstract Hardy algebra set in the context of a Fock space built from a $W^*$-correspondence E over a $W^{*}$-alge…
We define a new notion of entropy for operators on Fock spaces and positive definite multi-Toeplitz kernels on free semigroups. This is studied in connection with factorization theorems for (multi-Toeplitz, multi-analytic, etc.) operators on Fock spaces. These results lead to entropy inequalities and entropy formulas for positive definite multi-Toeplitz kernels on free semigroups (resp.multi-Toeplitz operators) and consequences concerning the extreme points of the unit ball of the noncommutative analytic Toeplitz algebra F_n^\infty. We obtain several geometric characterizations of the multivar…
We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let $\M$ be a von Neumann algebra equipped with a normal faithful semifinite trace $\t$, and let $E$ be an r.i. space on $(0, \8)$. Let $E(\M)$ be the associated symmetric space of measurable operators. Then to any bounded linear map $T$ from $E(\M)$ into a Hilbert space $\mathcal H$ corresponds a positive norm one functional $f\in E_{(2)}(\M)^*$ such that $$\forall x\in E(\M)\quad \|T(x)\|^2\le K^2 \|T\|^2 f(x^*x+xx^*),$$ where $E_{(2)}$ denotes the 2-concavification of $E$ and $K$ is a universal constan…
In this work, we solve the problem explicitly stated at the end of a paper of Junge, Mei and Parcet [JEMS2018, Problem C.5] for a large class of groups including all amenable groups and free groups. More precisely, we prove that the Hodge-Dirac operator of the canonical "hidden" noncommutative geometry associated with a Markov semigroup $(T_t)_{t \geq 0}$ of Fourier multipliers is bisectorial and admits a bounded $\mathrm{H}^\infty$ functional calculus on a bisector which implies a positive answer to the quoted problem. Our result can be seen as a strengthening of the dimension free estimates …
