Weighted Composition Operators on H 2 and Applications

Operators on function spaces acting by composition to the right with a fixed selfmap φ of some set are called composition operators of symbol φ. A weighted composition operator is an operator equal to a composition operator followed by a multiplication operator. We summarize the basic properties of bounded and compact weighted composition operators on the Hilbert Hardy space on the open unit disk and use them to study composition operators on Hardy–Smirnov spaces. Submitted: January 30, 2007. Revised: June 19, 2007. Accepted: July 11, 2007.     

Topological Structure of the Space of Weighted Composition Operators on H ∞

We study properties of the topological space of weighted composition operators on the space of bounded analytic functions on the open unit disk in the uniform operator topology. Moreover, we characterize the compactness of differences of two weighted composition operators.     

Applications of the Discrete Weiss Conjecture in Operator Theory

In this paper, we study a discrete version of the Weiss Conjecture. In Section 1 we discuss the Reproducing Kernel Thesis and in Section 2 we introduce the operators which concern us. Section 3 shows how to relate these operators to Carleson embeddings and weighted composition operators, so that we can apply the Carleson measure theorem to obtain conditions for boundedness and compactness of many weighted composition operators. Section 4 contains Theorem 4.4 which is a discrete version of the Weiss Conjecture for contraction semigroups, and finally Section 5 shows how the usual (continuous time) Weiss Conjecture is related to the discrete version studied here; in fact they are equivalent (for scalar valued observation operators). The main advantage of the discrete version is that it is technically simpler – the observation operators are automatically bounded and the functional calculus can be achieved using power series.     

Products of weighted composition and differentiation operators into weighted Zygmund and Bloch spaces

We characterize boundedness and compactness of products of differentiation operators and weighted composition operators between weighted Banach spaces of analytic functions and weighted Zygmund spaces or weighted Bloch spaces with general weights.     

Essential norm of the difference of weighted composition operators

We study differences of weighted composition operators between weighted Banach spaces H _ν ^∞ of analytic functions with weighted sup-norms and give an expression for the essential norm of these differences. We apply our result to estimate the essential norm of differences of composition operators acting on Bloch-type spaces. Authors’ addresses: Mikael Lindstr?m, Department of Mathematics, Abo Akademi University, FIN 20500 Abo, Finland; Elke Wolf, Mathematical Institute, University of Paderborn, D-33095 Paderborn, Germany     

Isometric Equivalence of Certain Operators on Banach Spaces

A pair of operators on a Banach space X are isometrically equivalent if they are intertwined by a surjective isometry of X. We investigate the isometric equivalence problem for pairs of operators on specific types of Banach spaces. We study weighted shifts on symmetric sequence spaces, elementary operators acting on an ideal I of Hilbert space operators, and composition operators on the Bloch space. This last case requires an extension of known results about surjective isometries of the Bloch space.     

Essential norm of differences of weighted composition operators between weighted-type spaces on the unit ball

We find an asymptotically equivalent expression to the essential norm of differences of weighted composition operators between weighted-type spaces of holomorphic functions on the unit ball in C^N. As a consequence we characterize the compactness of these operators. The boundedness of these operators is also characterized.     

Hermitian Weighted Composition Operators and Bergman Extremal Functions

Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper, Hermitian weighted composition operators on weighted Hardy spaces of the unit disk are studied. In particular, necessary conditions are provided for a weighted composition operator to be Hermitian on such spaces. On weighted Hardy spaces for which the kernel functions are ${(1 - \overline{w}z)^{-\kappa}}$ for κ ≥ 1, including the standard weight Bergman spaces, the Hermitian weighted composition operators are explicitly identified and their spectra and spectral decompositions are described. Some of these Hermitian operators are part of a family of closely related normal weighted composition operators. In addition, as a consequence of the properties of weighted composition operators, we compute the extremal functions for the subspaces associated with the usual atomic inner functions for these weighted Bergman spaces and we also get explicit formulas for the projections of the kernel functions on these subspaces.     

对称Fock空间上的加权复合算子

本文研究了可分Hilbert空间上的对称Fock空间上有界(线性和反线性)加权复合算子. 完全刻画了酉加权复合算子和自伴加权复合算子, 并考虑了一类正规加权复合算子.     

A note on weighted composition operators on the weighted Bergman space

This paper gives a note on weighted composition operators on the weighted Bergman space, which shows that for a fixed composition symbol, the weighted composition operators are bounded on the weighted Bergman space only with bounded weighted symbols if and only if the composition symbol is a finite Blaschke product.     

Toeplitz Operators on Arveson and Dirichlet Spaces

We define Toeplitz operators on all Dirichlet spaces on the unit ball of and develop their basic properties. We characterize bounded, compact, and Schatten-class Toeplitz operators with positive symbols in terms of Carleson measures and Berezin transforms. Our results naturally extend those known for weighted Bergman spaces, a special case applies to the Arveson space, and we recover the classical Hardy-space Toeplitz operators in a limiting case; thus we unify the theory of Toeplitz operators on all these spaces. We apply our operators to a characterization of bounded, compact, and Schatten-class weighted composition operators on weighted Bergman spaces of the ball. We lastly investigate some connections between Toeplitz and shift operators. The research of the second author is partially supported by a Fulbright grant.     

Differences of Composition Operators between Weighted Banach Spaces of Holomorphic Functions on the Unit Polydisk

We consider differences of composition operators between given weighted Banach spaces H ^∞ _v or H ⁰ _v of analytic functions defined on the unit polydisk D ^N with weighted sup-norms and give estimates for the distance of these differences to the space of compact operators. We also study boundedness and compactness of the operators. This paper is an extension of [6] where the one-dimensional case is treated. Received: May 15, 2007. Revised: October 8, 2007.     

Compact differences of composition operators

A characterization of compact difference is given for composition operators acting on the standard weighted Bergman spaces and necessary conditions are given on a larger scale of weighted Dirichlet spaces. Conditions are given under which a composition operator can be written as a finite sum of composition operators modulo the compacts. The additive structure of the space of composition operators modulo the compact operators is investigated further and a sufficient condition is given to insure that two composition operators lie in the same component.     

Little Hankel operators on the half plane

In this paper we consider a class of weighted integral operators onL ² (0, ) and show that they are unitarily equivalent to Hankel operators on weighted Bergman spaces of the right half plane. We discuss conditions for the Hankel integral operator to be finite rank, Hilbert-Schmidt, nuclear and compact, expressed in terms of the kernel of the integral operator. For a particular class of weights these operators are shown to be unitarily equivalent to little Hankel operators on weighted Bergman spaces of the disc, and the symbol correspondence is given. Finally the special case of the unweighted Bergman space is considered and for this case, motivated by approximation problems in systems theory, some asymptotic results on the singular values of Hankel integral operators are provided.     

Generalized Essential Norm of Weighted Composition Operators on some Uniform Algebras of Analytic Functions

We compute the essential norm of a composition operator relatively to the class of Dunford-Pettis operators or weakly compact operators, on some uniform algebras of analytic functions. Even in the context of H^∞ (resp. the disk algebra), this is new, as well for the polydisk algebras and the polyball algebras. This is a consequence of a general study of weighted composition operators.      

Product of Volterra Type Integral and Composition Operators on Weighted Fock Spaces

We characterize the bounded, compact, and Schatten class product of Volterra type integral and composition operators acting between weighted Fock spaces. Our results are expressed in terms of certain Berezin type integral transforms on the complex plane ?. We also estimate the norms and essential norms of these operators in terms of the integral transforms. All our results are valid for weighted composition operators when acting between the class of weighted Fock spaces considered.     

不同Bers型空间之间的加权复合算子

该文讨论了单位圆盘上不同Bers型空间之间的加权复合算子的有界性、紧性和弱紧性, 给出了一些充分必要的判别条件, 特别地得到不同Bers型空间上加权复合算子的紧性与弱紧性的等价性. 这些推广了经典的复合算子与乘法算子的相关结论. 该文同时也给出了Bers型空间上复合算子的Fredholm性和闭值域问题的刻画, 完善了文献[6]中结论.     

Fock空间上的一类加权复合算子

This paper studies a class of weighted composition operators and their spectrum on the Fock space.As an application,bounded self-adjoint,a class of complex symmetric weighted composition operators on the Fock space are characterized.     

Compact composition operators between weighted Bergman spaces on convex domains in ℂ n

It is shown that a compact composition operator on a weighted Bergman space over a smoothly bounded strongly convex domain in ⁿ can have no angular derivative. Also, sufficient conditions for the boundedness and the compactness of composition operators defined on Hardy and weighted Bergman spaces are obtained, for situations in which each of the target spaces is enlarged in a natural way.     

Norm-attaining weighted composition operators on weighted Banach spaces of analytic functions

We investigate weighted composition operators that attain their norm on weighted Banach spaces of holomorphic functions on the unit disc of type H ^∞. Applications for composition operators on weighted Bloch spaces are given.