Explore the vast range of titles available.

In this paper, we study matrix functions of bounded type from the viewpoint of describing an interplay between function theory and operator theory. We first establish a criterion on the coprime-ness of two singular inner functions and obtain several properties of the Douglas-Shapiro-Shields factorizations of matrix functions of bounded type. We propose a new notion of tensored-scalar singularity, and then answer questions on Hankel operators with matrix-valued bounded type symbols. We also examine an interpolation problem related to a certain functional equation on matrix functions of bounded type; this can be seen as an extension of the classical Hermite-Fejér Interpolation Problem for matrix rational functions. We then extend the

Truncated Toeplitz operators and their asymmetric versions are studied in the context of the Hardy space Hp of the half-plane for 1 < p < ∞. The question of uniqueness of the symbol is solved via the characterization of the zero operator. It is shown that asymmetric truncated Toeplitz operators are equivalent after extension to 2 × 2 matricial Toeplitz operators, which allows one to deduce criteria for Fredholmness and invertibility. Shifted model spaces are presented in the context of invariant subspaces, allowing one to derive new Beurling–Lax theorems.

In this paper, commutativity of the compressions of k th order slant Toeplitz operators is discussed. We show that the commutativity and essential commutativity of two compressions of k th order slant Toeplitz operators on H 2 are same. We also establish some results on the compactness of the compressions of k th order slant Toeplitz operators. In the last section, we discuss various similarities and differences between compactness and non-compactness of these operators by simply looking at their graphs.

This paper considers paired operators in the context of the Lebesgue Hilbert space on the unit circle and its subspace, the Hardy space H 2 . The kernels of such operators, together with their analytic projections, which are generalizations of Toeplitz kernels, are studied. Results on near-invariance properties, representations, and inclusion relations for these kernels are obtained. The existence of a minimal Toeplitz kernel containing any projected paired kernel and, more generally, any nearly S ∗ -invariant subspace of H 2 , is derived. The results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators.

In this paper, we characterize the hyponormal dual Toeplitz operators with special symbols on the orthogonal complement of the pluriharmonic Bergman space of the unit ball. Also we completely characterize the pluriharmonic symbols for (semi)commuting dual Toeplitz operators.

Suppose EE is a subset of the unit circle T\\mathbb {T} and H∞⊂L∞H^\\infty \\subset L^\\infty is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of EE to znH∞z^nH^\\infty. This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of certain conformal mappings.

We investigate some problems for truncated Toeplitz operators. Namely, the solvability of the Riccati operator equation on the set of all truncated Toeplitz operators on the model space Kθ = H2ΘθH2 is studied. We study in terms of Berezin symbols invertibility of model operators. We also prove some results for the Berezin number of the truncated Toeplitz operators. Moreover, we study some property for H2-functions in terms of noncyclicity of co-analytic Toeplitz operators and hypercyclicity of model operators.

We present two novel results about Hilbert space operators which are nilpotent of order two. First, we prove that such operators are indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a complex symmetric operator. In fact, we prove that this property characterizes nilpotents of order two among all nonzero bounded operators. Second, we establish that every nilpotent of order two is unitarily equivalent to a truncated Toeplitz operator.

This work explores joint hyponormality of Toeplitz pairs. Topics include: hyponormality of Toeplitz pairs with one co-ordinate a Toeplitz operator with analytic polynomial symbol; hyponormality of trigonometric Toeplitz pairs; and the gap between $2$-hyponormality and subnormality.

High level academic ebook