A HYPONORMAL TOEPLITZ COMPLETION PROBLEM

In this paper, we investigate properties of hyponormal Toeplitz operators whose symbols are analytic or co-analytic on a Newton space. We establish both necessary and sufficient conditions for a Toeplitz operator Tφ to be hyponormal or normal. Additionally, we give some applications of such results.

In this article, we introduce a new class of conjugations in the scalar-valued Hardy space and provide a characterization of a complex symmetric Toeplitz operator with respect to these newly introduced conjugations in various cases. Moreover, we obtain a characterization of a complex symmetric block Toeplitz operator on the vector-valued Hardy space with respect to certain conjugations introduced in [Câmara MC, Kliś-Garlicka K, Ptak M. Complex symmetric completions of partial operator matrices. Linear and Multilinear Algebra. 2019; DOI: 10.1080/03081087.2019.1631246], [Kang D, Ko E, Lee JE. Remarks on complex symmetric Toeplitz operators. Linear Multilinear Algebra. 2020; DOI: 10.1080/03081087.2020.1842847], [Ko E, Lee JE. Remark on complex symmetric operator matrices. Linear Multilinear Algebra. 2019;67(6):1198–1216].

Toeplitz and asymptotic Toeplitz operators on [formula omitted

In this work we consider the hyponormality of Toeplitz operators on the Bergman space of the annulus with a logarithmic weight. We give necessary conditions when the symbol is of the form ψ + φ where ψ and φ are analytic on the annulus {z ∈ C,1/2 < |z| < 1} .

A bounded Hilbert space operatorTis hyponormal ifT∗T−TT∗is a positive operator. We consider the hyponormality of Toeplitz operators on a weighted Bergman space. We find a necessary condition for hyponormality in the case of a symbol of the formf+g¯wherefandgare bounded analytic functions on the unit disk. We then find sufficient conditions whenfis a monomial.

In this paper we consider the hyponormality of Toeplitz operators <TEX>$T_{\varphi}$</TEX> on the Bergman space <TEX>$L_a^2{(\mathbb{D})$</TEX> in the cases, where <TEX>${\varphi}\;:=f+\bar{g}$</TEX> (f and g are polynomials). We present some necessary or sufficient conditions for the hyponormality of <TEX>$T_{\varphi}$</TEX> under certain assumptions about the coefficients of <TEX>${\varphi}$</TEX>.

In this note we consider the hyponormality of Toeplitz operators <TEX>$T_{\varphi}$</TEX> on the Weighted Bergman space <TEX>$A^2_{\alpha}(\mathbb{D})$</TEX> with symbol in the class of functions <TEX>$f+\bar{g}$</TEX> with polynomials <TEX>$f$</TEX> and <TEX>$g$</TEX> of degree 2.

We derive an asymptotic formula for the argument of a Blaschke product in the upper half-plane with purely imaginary zeros. We then use this formula to find conditions for the quotient of two such Blaschke products to be continuous on the real line. These results are applied to certain Hankel and Toeplitz operators arising in the Cauchy problem for the Korteweg-de Vries equation. Our main theorems include certain compactness conditions for Hankel operators and invertibility conditions for Toeplitz operators with oscillating symbols containing such quotients. As a by-product we obtain a well-posedness result for the Korteweg-de Vries equation.

It is shown that rational functions continuous on the unit circle and of maximal equios-cillation order are either nonconstant finite Blaschke products or nonconstant inverted finite Blaschke products.

Explicit formulas for the (generalized) inverse and criteria of invertibility are given for block Toeplitz and Wiener-Hopf type operators. We consider operators with symbols defined on a curve composed of several non-intersecting simple closed contours. Also criteria and explicit formulas for canonical factorization of matrix functions relative to a compound contour are presented. The matrix functions we work with are rational on each of the compounding contours but the rational expressions may vary from contour to contour. We use realizations for each of the rational expressions and the final results are stated in terms of invertibility properties of a certain finite matrix called indicator, which is built from the realizations. The analysis does not depend on finite dimensionality and is carried out for operator valued symbols.

The little Bloch space, B0, is the space of all holomorphic functions f on the unit disk such that limlzl?1 lf'(z)l (1- lzl2) = 0. Finite Blaschke products are clearly in B0, but examples of infinite products in B0 are more difficult to obtain (there are now several constructions due to Sarason, Stephenson and the author, among others). Stephenson has asked whether B0 contains an infinite, indestructible Blaschke product, i.e., a Blaschke product B so that (B(z) - a)/(1 - âB(z)), is also a Blaschke product for every element a I D. In this paper we give an afirmative answer to his question by constructing such a Blaschke product. We also answer a question of Carmona and Cufi by constructing a VMO function, f, so that ll f ll8 = 1 and whose range set, R(f,a) = {w : there exists zn ? a, f(zn) = w}, equals the open unit disk for every a I T.

A symmetric matrix $C$ is completely positive (CP) if there exists an entrywise nonnegative matrix $B$ such that $C=BB^T$. The CP-completion problem is to study whether we can assign values to the missing entries of a partial matrix (i.e., a matrix having unknown entries) such that the completed matrix is completely positive. We propose a semidefinite algorithm for solving general CP-completion problems and study its properties. When all of the diagonal entries are given, the algorithm can give a certificate if a partial matrix is not CP-completable, and it almost always gives a CP-completion if it is CP-completable. When diagonal entries are partially given, similar properties hold. Computational experiments are also presented to show how CP-completion problems can be solved.

This paper is a survey of some works on paracommutators and Hankel operators in China, which are the subjects I have been studying since 1985. The classical Hankel operator is defined on the Hilbert space l 2 by the Hankel matrix Γ = {b n+k } n,k ≥0, and the Toeplitz operator is defined also on l 2 by the toeplitz matrix T = {b n−k } n≥k≥0. By the theory of the Fourier series both of them become the operators on the Hardy space \( {{H}^{2}}(T) = \{ f \in {{L}^{2}}(T):\hat{f}(n) = 0,n < 0\} . \). Let \( \overline H ^2 (T) \) denote the conjugate Hardy space and P,P denote the ortnogonal projections from L 2(T) into H 2(T),H 2 (T) respectively. Then the Toeplitz operator is defined as follows $$ {T_b}(f) = P(\bar bf),for f \in {H^2}(T) $$ (1.1) and the Hankel operator is defined as follows $$ {H_b}(f) = \bar P(\bar bf),for f \in {H^2}(T).$$ (1.2)

The symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit disc in several aspects. This space is known as the Hardy space of the symmetrized bidisc. We introduce the study of those operators on the Hardy space of the symmetrized bidisc that are analogous to Toeplitz operators on the Hardy space of the unit disc. More explicitly, we first study multiplication operators on a bigger space (an $L^2$-space) and then study compressions of these multiplication operators to the Hardy space of the symmetrized bidisc and prove the following major results. (1) Theorem I analyzes the Hardy space of the symmetrized bidisc, not just as a Hilbert space, but as a Hilbert module over the polynomial ring and finds three isomorphic copies of it as $\mathbb D^2$-contractive Hilbert modules. (2) Theorem II provides an algebraic, Brown and Halmos-type characterization of Toeplitz operators. (3) Theorem III gives several characterizations of an analytic Toeplitz operator. (4) Theorem IV characterizes asymptotic Toeplitz operators. (5) Theorem V is a commutant lifting theorem. (6) Theorem VI yields an algebraic characterization of dual Toeplitz operators. Every section from Section 2 to Section 7 contains a theorem each, the main result of that section.

Consider the multiplication operator MB in $L^2(\mathbb{T})$, where the symbol B is a finite Blaschke product. In this article, we characterize the commutant of MB in $L^2(\mathbb{T})$. As an application of this characterization result, we explicitly determine the class of conjugations commuting with $M_{z^2}$ or making $M_{z^2}$ complex symmetric by introducing a new class of conjugations in $L^2(\mathbb{T})$. Moreover, we analyse their properties while keeping the whole Hardy space, model space and Beurling-type subspaces invariant. Furthermore, we extended our study concerning conjugations in the case of finite Blaschke products.