Schur Agler Class Research Articles

Spectra of some weighted composition operators on the ball

We find sufficient conditions for a self-map of the unit ball to converge uniformly under iteration to a fixed point or idempotent on the entire ball. Using these tools, we establish spectral containments for weighted composition operators on Hardy and Bergman spaces of the ball. When the compositional symbol is in the Schur–Agler class, we establish the spectral radii of these weighted composition operators.

A polynomial approximation result for free Herglotz–Agler functions

AbstractIn this paper, we prove a noncommutative (nc) analog of Schwarz lemma for the nc Schur–Agler class and prove that the regular nc Schur–Agler class and the regular free Herglotz–Agler class are homeomorphic. Moreover, we give a characterization of regular free Herglotz–Agler functions. As an application, we will show that any regular free Herglotz–Agler functions can uniformly be approximated by regular Herglotz–Agler free polynomials.

Factorizations of Schur functions

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 $$\begin{aligned} {\mathcal {S}}({\mathbb {D}}) = \left\{ \varphi \in H^\infty ({\mathbb {D}}): \Vert \varphi \Vert _{\infty } := \sup _{z \in {\mathbb {D}}} |\varphi (z)| \le 1\right\} . \end{aligned}$$The elements of \({\mathcal {S}}({\mathbb {D}})\) are called Schur functions. A classical result going back to I. Schur states: A function \(\varphi : {\mathbb {D}} \rightarrow {\mathbb {C}}\) is in \({\mathcal {S}}({\mathbb {D}})\) if and only if there exist a Hilbert space \({\mathcal {H}}\) and an isometry (known as colligation operator matrix or scattering operator matrix) $$\begin{aligned} V = \begin{bmatrix} a &{}\quad B \\ C &{}\quad D \end{bmatrix} : {\mathbb {C}} \oplus {\mathcal {H}} \rightarrow {\mathbb {C}} \oplus {\mathcal {H}}, \end{aligned}$$such that \(\varphi \) admits a transfer function realization corresponding to V, that is $$\begin{aligned} \varphi (z) = a + z B (I_{{\mathcal {H}}} - z D)^{-1} C \quad \quad (z \in {\mathbb {D}}). \end{aligned}$$An analogous statement holds true for Schur functions on the bidisc. On the other hand, Schur-Agler class functions on the unit polydisc in \({\mathbb {C}}^n\) is a well-known “analogue” of Schur functions on \({\mathbb {D}}\). In this paper, we present algorithms to factorize Schur functions and Schur-Agler class functions in terms of colligation matrices. More precisely, we isolate checkable conditions on colligation matrices that ensure the existence of Schur (Schur-Agler class) factors of a Schur (Schur-Agler class) function and vice versa.

Isometric dilations and von Neumann inequality for finite rank commuting contractions

Motivated by Ball, Li, Timotin and Trent's Schur-Agler class version of commutant lifting theorem, we introduce a class, denoted by Pn(H), of n-tuples of commuting contractions on a Hilbert space H. We always assume that n≥3. The importance of this class of n-tuples stems from the fact that the von Neumann inequality or the existence of isometric dilation does not hold in general for n-tuples, n≥3, of commuting contractions on Hilbert spaces (even in the level of finite dimensional Hilbert spaces). Under some rank-finiteness assumptions, we prove that tuples in Pn(H) always admit explicit isometric dilations and satisfy a refined von Neumann inequality in terms of algebraic varieties in the closure of the unit polydisc in Cn.

On the Nevanlinna problem: characterization of all Schur–Agler class solutions affiliated with a given kernel

Given a domain $\Omega$ in $\mathbb{C}^m$, and a finite set of points $z_1,z_2,\ldots, z_n\in \Omega$ and $w_1,w_2,\ldots, w_n\in \mathbb{D}$ (the open unit disc in the complex plane), the \textit{Pick interpolation problem} asks when there is a holomorphic function $f:\Omega \rightarrow \overline{\mathbb{D}}$ 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 $\Omega=\mathbb{D}$. Nevanlinna characterized all possible functions $f$ that \textit{interpolate} the data. We generalize Nevanlinna's result to a domain $\Omega$ in $\mathbb{C}^m$ admitting holomorphic test functions when the function $f$ comes from the Schur-Agler class and is affiliated with a certain completely positive kernel. The Schur class is a naturally associated Banach algebra of functions with a domain. The success of the theory lies in characterizing the Schur class interpolating functions for three domains - the bidisc, the symmetrized bidisc and the annulus - which are affiliated to given kernels.

Scattering Systems with Several Evolutions and Formal Reproducing Kernel Hilbert Spaces

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 linear system. Previous work of the authors identified the Schur–Agler class as those Schur-class functions which arise as the scattering matrix for a certain type of (not necessarily minimal) Lax–Phillips multievolution scattering system having some additional geometric structure. The present paper links this additional geometric scattering structure directly with a known reproducing-kernel characterization of the Schur–Agler class. We use extensively the technique of formal reproducing kernel Hilbert spaces that was previously introduced by the authors and that allows us to manipulate formal power series in several commuting variables and their inverses (e.g., Fourier series of elements of \(L^2\) on a torus) in the same way as one manipulates analytic functions in the usual setting of reproducing kernel Hilbert spaces.

Norm-Constrained Determinantal Representations of Multivariable Polynomials

For every multivariable polynomial \(p\), with \(p(0)=1\), we construct a determinantal representation, \( p=\det (I - K Z )\), where \(Z\) is a diagonal matrix with coordinate variables on the diagonal and \(K\) is a complex square matrix. Such a representation is equivalent to the existence of \(K\) whose principal minors satisfy certain linear relations. When norm constraints on \(K\) are imposed, we give connections to the multivariable von Neumann inequality, Agler denominators, and stability. We show that if a multivariable polynomial \(q\), \(q(0)=0,\) satisfies the von Neumann inequality, then \(1-q\) admits a determinantal representation with \(K\) a contraction. On the other hand, every determinantal representation with a contractive \(K\) gives rise to a rational inner function in the Schur–Agler class.

High-order Schwarz-Pick Lemma for the Schur class on the polydisc

In this paper, we give Schwarz-Pick Lemma for functions in the Schur class on the polydisc of ℂn, and generalize some early work of Schwarz-Pick Lemma for functions in the Schur class on the unit disk of C and functions in the Schur-Agler class on the polydisc of ℂn.

Test Functions, Schur–Agler Classes and Transfer-Function Realizations: The Matrix-Valued Setting

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 algebra (bounded analytic functions on the unit disk with derivative at the origin constrained to have zero value). Emphasis is on examples where the matrix-valued version is not obtained as a simple tensoring with $${{\mathbb C}^{N}}$$ of the scalar-valued version.

Rational inner functions in the Schur-Agler class of the polydisk

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 suggest.

Schur-Agler class rational inner functions on the tridisk

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 ) (n,1,1) is in the Schur-Agler class, and every rational inner function of degree ( n , m , 1 ) (n,m,1) is in the Schur-Agler class after multiplication by a monomial of sufficiently high degree.

Stable symmetric polynomials and the Schur–Agler class

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.

Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality

We obtain a decomposition for multivariable Schur-class functions on the unit polydisk which, to a certain extent, is analogous to Agler's decomposition for functions from the Schur–Agler class. As a consequence, we show that d-tuples of commuting strict contractions obeying an additional positivity constraint satisfy the d-variable von Neumann inequality for an arbitrary operator-valued bounded analytic function on the polydisk. Also, this decomposition yields a necessary condition for solvability of the finite data Nevanlinna–Pick interpolation problem in the Schur class on the unit polydisk.

Norms and spectral radii of linear fractional composition operators on the ball

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 new 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.

Schwarz-Pick Inequalities for the Schur-Agler Class on the Polydisk and Unit Ball

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.

A Schwarz lemma on the polydisk

We prove a generalization of the infinitesimal portion of the classical Schwarz lemma for functions from the polydisk to the disk. In particular, we describe the functions which play the role of automorphisms of the disk in this context–they turn out to be rational inner functions in the Schur-Agler class of the polydisk with an added symmetry constraint. In addition, some sufficient conditions are given for a function to be of this type.

Interpolation in semigroupoid algebras

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 Carathéodory-Fejér interpolation problems and their multivariable commutative and noncommutative generalizations, this approach also covers more exotic problems.

Carathéodory interpolation on the non-commutative polydisk

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 × n matrices, for all n ∈ N .

Realization and interpolation for Schur–Agler-class functions on domains with matrix polynomial defining function in [formula omitted

We consider a bitangential interpolation problem for operator-valued functions defined on a general class of domains in C n (including as particular cases, Cartan domains of types I–III) which satisfy a type of von Neumann inequality associated with the domain. We show that any such function has a realization in terms of a unitary colligation and the defining polynomial for the domain. We show how the solution of various classes of bitangential interpolation problems for this class of functions corresponds to a unitary extension of a particular partially defined isometry uniquely specified by the interpolation data. Criteria for existence of solutions are given (1) in terms of positivity of a certain kernel completely determined by the data, or, more generally, (2) by the existence of a positive-kernel solution of a certain generalized Stein equation completely determined by the data.

A tangential interpolation problem on the distinguished boundary of the polydisk for the Schur–Agler class

We study a tangential interpolation problem with an arbitrary set of interpolating points on the distinguished boundary of the unit polydisk for Schur–Agler class. The description of all solutions is parametrized in terms of a linear fractional transformation.