Contractive Analytic Functions Research Articles

Spectral measures for derivative powers via matrix-valued Clark theory

The theory of finite-rank perturbations allows for the determination of spectral information for broad classes of operators using the tools of analytic function theory. In this work, finite-rank perturbations are applied to powers of the derivative operator, providing a full account from self-adjoint boundary conditions to computing aspects of the operators' matrix-valued spectral measures. In particular, the support and weights of the Clark (spectral) measures are computed via the connection between matrix-valued contractive analytic functions and matrix-valued nonnegative measures through the Herglotz Representation Theorem. For operators associated with several powers of the derivative, explicit formulae for these measures are included. While eigenfunctions and eigenvalues for these operators with fixed boundary conditions can often be computed using direct methods from ordinary differential equations, this approach provides a more complete picture of the spectral information.

Distinguished varieties through the Berger-Coburn-Lebow theorem

A distinguished algebraic variety in $\mathbb{C}^2$ has been the focus of much research in recent years because of good reasons. This note gives a different perspective. (1) We find a new characterization of an algebraic variety $\mathcal W$ which is distinguished with respect to the bidisc. It is in terms of the joint spectrum of a pair of commuting linear matrix pencils. (2) There is a characterization known of $\mathbb{D}^2\cap\mathcal{W}$ due to a seminal work of Agler and McCarthy. We show that Agler--McCarthy characterization can be obtained from the new one and vice versa. (3) En route, we develop a new realization formula for operator-valued contractive analytic functions on the unit disc. (4) There is a one-to-one correspondence between operator valued contractive holomorphic functions and {\em canonical model triples}. This pertains to the new realization formula mentioned above. (5) Pal and Shalit gave a characterization of an algebraic variety, which is distinguished with respect to the symmetrized bidisc, in terms of a matrix of numerical radius no larger than $1$. We refine their result by making the class of matrices strictly smaller. (6) In a generalization in the direction of more than two variables, we characterize all one-dimensional algebraic varieties which are distinguished with respect to the polydisc. At the root of our work is the Berger--Coburn--Lebow theorem characterizing a commuting tuple of isometries.

Aleksandrov–Clark Theory for Drury–Arveson Space

Recent work has demonstrated that Clark’s theory of unitary perturbations of the backward shift on a deBranges–Rovnyak space on the disk has a natural extension to the several-variable setting. In the several-variable case, the appropriate generalization of the Schur class of contractive analytic functions is the closed unit ball of the Drury–Arveson multiplier algebra and the Aleksandrov–Clark measures are necessarily promoted to positive linear functionals on a symmetrized subsystem of the Free Disk operator system $$\mathcal {A} _d + \mathcal {A} _d ^*$$ , where $$\mathcal {A} _d$$ is the Free or Non-commutative Disk Algebra on d generators. We continue this program for vector-valued Drury–Arveson space by establishing the existence of a canonical ‘tight’ extension of any Aleksandrov–Clark map to the full Free Disk operator system. We apply this tight extension to generalize several earlier results and we characterize all extensions of the Aleksandrov–Clark maps.

Harnack and Shmul’yan pre-order relations for Hilbert space contractions

We study the behavior of some classes of Hilbert space contractions with respect to Harnack and Shmul’yan pre-orders and the corresponding equivalence relations. We give some conditions under which the Harnack equivalence of two given contractions is equivalent to their Shmul’yan equivalence and to the existence of an arc joining the two contractions in the class of operator-valued contractive analytic functions on the unit disc. We apply some of these results to quasi-isometries and quasi-normal contractions, as well as to partial isometries for which we show that their Harnack and Shmul’yan parts coincide. We also discuss an extension, recently considered by S. ter Horst (2014), of the Shmul’yan pre-order from contractions to the operator-valued Schur class of functions. In particular, the Shmul’yan–ter Horst part of a given partial isometry, viewed as a constant Schur class function, is explicitly determined.

On a Local Darlington Synthesis Problem

The Darlington synthesis problem (in the scalar case) is a problem of embedding a given contractive analytic function to an inner \(2\times 2\) matrix function as an entry. A fundamental result of Arov–Douglas–Helton relates this algebraic property to a purely analytic one known as a pseudocontinuation of bounded type. We suggest a local version of the Darlington synthesis problem and prove a local analog of the ADH theorem.

Does Every Contractive Analytic Function in a Polydisk have a Dissipative n-Dimensional Scattering Realization?

The answer to the title question is “no.” The Kaijser–Varopoulos counterexample to the three-variable von Neumann inequality shows that conservative realizations do not always exist; in this note we show that this same example can be used to prove that dissipative realizations need not exist. (This is a consequence of the particular features of this example; it need not be true of every counterexample to von Neumann’s inequality.)

On metric properties of limit sets of contractive analytic non-Archimedean dynamical systems

Let K be an algebraically closed field which is complete with respect to a non-trivial non-Archimedean absolute value |⋅|. We study metric properties of the limit set Λ of a semigroup G generated by a finite set of contractive analytic functions on O={z∈K||z|⩽1}. We prove that the limit set Λ of G is uniformly perfect if the derivative of each generating function of G does not vanish on O. Furthermore, we show that if each coefficient of the generating functions is in the field Qp of p-adic numbers, or the limit set Λ satisfies the strong open set condition, then Λ has the doubling property. This yields that the limit set Λ is quasisymmetrically equivalent to the space Z2 of 2-adic integers. We also give a counterexample to show that not all limit sets have the doubling property. The Berkovich space is introduced to study the limit set Λ, and we prove that the limit set Λ has a positive capacity in the Berkovich space which yields that there exists an equilibrium measure μ whose support is contained in the limit set Λ. We also show that if the semigroup is generated by a countable set of contractive analytic functions, then its limit set Λ can be non-compact. However, if coefficients of the generating functions lie in Qp, then the limit set Λ is compact.

Unitary Perturbations of Compressed n-Dimensional Shifts

Given any contractive matrix-valued analytic function $${\Theta}$$ on the unit disc $${\mathbb{D}}$$ , we construct a $${\mathcal{U}(n)}$$ -parameter family of unitary operators which correspond to $${\Theta}$$ in a natural way. These operators are unitarily equivalent to higher dimensional analogues of Clark’s unitary perturbations, a family of unitary operators associated to any self-map of the unit disc. Clark’s unitary perturbations were introduced in a seminal paper of Clark which has inspired the study of what are now called Aleksandrov–Clark measures. Our higher dimensional analogues of Clark’s unitary perturbations are applied to obtain matrix-generalizations of several classical results on the Aleksandrov–Clark measures associated to any holomorphic self-map of the unit disc. In particular we establish a matrix-valued Aleksandrov disintegration theorem for the Aleksandrov–Clark measures associated with matrix-valued contractive analytic functions $${\Theta}$$ , and, by following results of Clark and Fricain in the scalar case, we provide a necessary and sufficient condition for the de Branges–Rovnyak space associated with $${\Theta}$$ to contain a total orthogonal set of point evaluation vectors.

The analytic structure of the reflection coefficient, a sum rule and a complete description of the Weyl m-function of half-line Schrödinger operators with L2-type potentials

We prove that the reflection coefficient of one-dimensional Schrödinger operators with potentials supported on a half-line can be represented in the upper half-plane as the quotient of a contractive analytic function and a properly regularized Blaschke product. We apply this fact to obtain a new trace formula and trace inequality for the reflection coefficient that yields a description of the Weyl m-function of Dirichlet half-line Schrödinger operators with slowly decaying potentials q subject to Among others, we also refine the 3/2-Lieb-Thirring inequality.

The Arov?Grossman Model and the Burg Multivariate Entropy

In this article we deal with the Arov–Grossman functional model to describe all the solutions of the Covariance Extension Problem for q-variate stationary stochastic processes and we find the density that maximizes the Burg Multivariate Entropy. This description is based on a one-to-one correspondence between the set of all solutions of the Covariance Extension Problem and the set of all contractive analytic functions H from the open unit disk with values on the space of q × q matrices. With this correspondence, the density that maximizes the Burg Multivariate Entropy corresponds to the function H\equiv0. Also, from the information that the Arov–Grossman functional model provides we obtain a version of the Levinson algorithm. The partial autocorrelation coefficient matrices are computed directly from Levinson’s recursions.

Bergman-Type Reproducing Kernels, Contractive Divisors, and Dilations

Let Ω be a region in the complex plane. In this paper we introduce a class of sesquianalytic reproducing kernels on Ω that we call B-kernels. When Ω is the open unit disk D and certain natural additional hypotheses are added we call such kernels k Bergman-type kernels. In this case the associated reproducing kernel Hilbert space H(k) shares certain properties with the classical Bergman space L2α of the unit disk. For example, the weighted Bergman kernels kβw(z)=(1−wz)−β, 1⩽β⩽2 are Bergman-type kernels. Furthermore, for any Bergman-type kernel k one has H2⊆H(k)⊆L2a, where the inclusion maps are contractive, and Mζ, the operator of multiplication with the identity function ζ, defines a contraction operator on H(k). Our main results about Bergman-type kernels k are the following two: First, once properly normalized, the reproducing kernel for any nontrivial zero based invariant subspace M of H(k) is a Bergman-type kernel as well. For the weighted Bergman kernels kβ this result even holds for all Mζ-invariant subspace M of index 1, i.e., whenever the dimension of M/ζM is one. Second, if M is any multiplier invariant subspace of H(k), and if we set C∗=M⊖zM, then Mζ∣M is unitarily equivalent to Mζ acting on a space of C∗-valued analytic functions with an operator-valued reproducing kernel of the typekw(z)=(IC∗−zwV(z)V(w)*)kw(z), where V is a contractive analytic function V:D→L(E, C∗), for some auxiliary Hilbert space E. Parts of these theorems hold in more generality. Corollaries include contractive divisor, wandering subspace, and dilation theorems for all Bergman-type reproducing kernel Hilbert spaces. When restricted to index one invariant subspaces of H(kβ), 1⩽β⩽2, our approach yields new proofs of the contractive divisor property, the strong contractive divisor property, and the wandering subspace theorems and inner–outer factorization. Our proofs are based on the properties of reproducing kernels, and they do not involve the use of biharmonic Green functions as had some of the earlier proofs.