Function Theoretic Characterization Research Articles

Characterizations for the difference of weighted composition operators over the polydisk

In this paper, we have established the characterizations for the boundedness and compactness of the difference of weighted composition operators acting on the weighted Bergman spaces over the polydisk when weight functions are Lebesgue integrable. In particular, our results contain the Reproducing kernel thesis for the difference of weighted composition operators. As an application, we obtain a function-theoretic characterization when the weight functions are holomorphic and essentially bounded.

The Toeplitzness of weighted composition operators on Banach spaces of holomorphic functions

ABSTRACT The concepts of various kinds of asymptotically Toeplitz operators have been originally defined and studied for the Hardy–Hilbert space. The aim of this article is to extend these definitions to the case of operators acting on Banach spaces of holomorphic functions, including Hardy spaces, Hardy–Lorentz spaces, and Hardy–Orlicz spaces. We give a function-theoretic characterization of uniformly asymptotically Toeplitzness and mean weakly asymptotically Toeplitzness for weighted composition operators. We also investigate strong and weak asymptotic Toeplitzness of weighted composition operators.

Function theoretic characterizations of Weil–Petersson curves

The Weil–Petersson class is the closure of the smooth closed curves in the Weil–Petersson metric on universal Teichmüller space defined by Takhtajan and Teo. We give some new characterizations of this class of curves and some new proofs of previously known characterizations. In particular, we give a new, more geometric characterization of the conformal weldings of such curves and characterize the curves themselves in terms of Peter Jones’s \beta -numbers.

Compactness and weak compactness of weighted composition operators on BMOA

We show that a weighted composition operator W ψ , φ {W_{\psi ,\varphi }} is compact on BMOA precisely when it is unconditionally converging. As a direct consequence to this result we deduce that all weakly compact or completely continuous weighted composition operators on BMOA are compact. The proof of this result is based on a new simplified function-theoretic characterization of the compactness of weighted composition operators on BMOA.

Product-Type Operators on the Space of Fractional Cauchy Transforms

The space of fractional Cauchy transforms plays a central role in classical complex analysis, harmonic analysis, and geometric measure theory. In this paper, we study the boundedness and compactness of product-type operators from the space of fractional Cauchy transforms to the Zygmund-type space in terms of the function theoretic characterization of Julia–Carathéodory type.

Difference of composition operators on spaces of vector-valued holomorphic functions

In this paper, we completely characterize the boundedness of difference of weighted composition operators between weak and strong vector-valued Bergman spaces in three terms: one is a function theoretic characterization of Julia-Carathéodory type, the second is a power type characterization and the other is a measure theoretic characterization of Carleson type. Furthermore, the bounded difference of composition operators is investigated for corresponding vector-valued Fock space case, which is in sharp contrast with some phenomenon on the setting of vector-valued Bergman space.

Compact difference of composition operators with smooth symbols on the unit ball

A long-standing problem raised by Shapiro and Sundberg in 1990, the characterization for compact differences of composition operators acting on the Hardy space over the unit disk, has been recently obtained in terms of certain Carleson measures ([4]). The function theoretic characterization for compact differences still remains open in the Hardy space case even on the unit disc and the situation is much more complicated for the several variables case. In this article, we investigate a function theoretic characterization of the compact difference on the Hardy or the Bergman spaces over the unit ball. The condition ρ(φ(z),ψ(z))→0 as max⁡{|φ(z)|,|ψ(z)|}→1 is shown to be sufficient for the difference, Cφ−Cψ, of two composition operators to be compact on the Hardy or the Bergman spaces over the unit ball when each single composition operator is bounded. We show that this condition is also a necessary condition if the symbols are of class Lip1(B).

Compact linear combination of composition operators on Bergman spaces

Motivated by the question of Shapiro and Sundberg raised in 1990, study on linear combinations of composition operators has been a topic of growing interest. In this paper, we completely characterize the compactness of any finite linear combination of composition operators with general symbols on the weighted Bergman spaces in two classical terms: one is a function theoretic characterization of Julia-Carathéodory type and the other is a measure theoretic characterization of Carleson type. Our approach is completely different from what has been known so far.

Difference of Weighted Composition Operators on the Space of Cauchy Integral Transforms

In this paper, we provide a complete function theoretic characterizations for boundedness and compactness of difference of weighted composition operators from the space of Cauchy integral transforms to logarithmic weighted-type spaces. Surprisingly, an interesting feature of these characterizations is that they are free from pseudo-hyperbolic distance between $\varphi(z)$ and $\psi(z)$, which is different from the previous characterizations of difference of weighted composition operators acting between different holomorphic function spaces.

Intersection of harmonically weighted Dirichlet spaces

In 1991, S. Richter introduced harmonically weighted Dirichlet spaces D(μ), motivated by his study of cyclic analytic two-isometries. In this paper, we consider ⋂μ∈PD(μ), the intersection of D(μ) spaces, where P is the family of Borel probability measures. Several function-theoretic characterizations of the Banach space ⋂μ∈PD(μ) are given. We also show that ⋂μ∈PD(μ) is located strictly between some classical analytic Lipschitz spaces and ⋂μ∈PD(μ) can be regarded as the endpoint case of analytic Morrey spaces in some sense.

Extensions of symmetric operators I: The inner characteristic function case

AbstractGiven a symmetric linear transformation on a Hilbert space, a natural problem to consider is the characterization of its set of symmetric extensions. This problem is equivalent to the study of the partial isometric extensions of a fixed partial isometry. We provide a new function theoretic characterization of the set of all self-adjoint extensions of any symmetric linear transformation B with finite equal indices and inner Livšic characteristic function θ

Compact composition operators acting between weighted Bergman spaces of the unit ball

In this paper, we study a composition operator \({C_{\varphi}}\) on the weighted Bergman space \({A_{\alpha}^p(B)}\) of the unit ball B in \({{\mathbb{C}}^N}\) . Under a natural condition we estimate the essential norm of \({C_{\varphi}}\) . As a consequence of this estimate, we also give a function-theoretic characterization of \({\varphi}\) that induces a compact composition operator on \({A_{\alpha}^p(B)}\) .

On the Intertwining of $$\partial{\mathcal{D}}$$ -Isometries

Let \({\mathcal{D}}\) be a strictly pseudoconvex bounded domain in \({\mathbb{C}}^m\) with C2 boundary \(\partial{\mathcal{D}}\). If a subnormal m-tuple T of Hilbert space operators has the spectral measure of its minimal normal extension N supported on \(\partial{\mathcal{D}}\), then T is referred to as a \(\partial{\mathcal{D}}\)-isometry. Using some non-trivial approximation theorems in the theory of several complex variables, we establish a commutant lifting theorem for those \(\partial{\mathcal{D}}\)-isometries whose (joint) Taylor spectra are contained in a special superdomain Ω of \({\mathcal{D}}\). Further, we provide a function-theoretic characterization of those subnormal tuples whose Taylor spectra are contained in Ω and that are quasisimilar to a certain (fixed) \(\partial{\mathcal{D}}\)-isometry T (of which the multiplication tuple on the Hardy space of the unit ball in \({\mathbb{C}}^m\) is a rather special example).

Composition Operators and Vector-valued BMOA

Analytic composition operators \(C_{\varphi}: f \mapsto f \, o\, \varphi\) are studied on X-valued versions of BMOA, the space of analytic functions on the unit disk that have bounded mean oscillation on the unit circle, where X is a complex Banach space. It is shown that if X is reflexive and Cφ is compact on BMOA, then Cφ is weakly compact on the X-valued space BMOAC(X) defined in terms of Carleson measures. A related function-theoretic characterization is given of the compact composition operators on BMOA.

Composition Operators on 𝑄𝑝 Spaces

Suppose that is a nonconstant analytic self-mapping of the unit disk 𝐷. Necessary or sufficient conditions for the composition operator 𝐶 : 𝑄𝑞 𝑄𝑝 to be bounded or compact are given in terms of . The function-theoretic characterization of the compactness of the composition operator 𝐶 : 𝑄𝑞 𝑄𝑝,0 is also given.

Composition Operators on Bergman Spaces

The authors obtain function theoretic characterizations of the compactness on the standard weighted Bergman spaces of the two operators formed by multiplying a composition operator with the adjoint of another composition operator.

Isolated points and essential components of composition operators on 𝐻^{∞}

We consider the topological space of all composition operators on the Banach algebra of bounded analytic functions on the unit disk. We obtain a function theoretic characterization of isolated points and show that each isolated composition operator is essentially isolated.

Compactness of composition operators on BMOA

A function theoretic characterization is given of when a composition operator is compact on BMOA, the space of analytic functions on the unit disk having radial limits that are of bounded mean oscillation on the unit circle. When the symbol of the composition operator is univalent, compactness on BMOA is shown to be equivalent to compactness on the Bloch space, and a characterization in terms of the geometry of the image of the disk under the symbol of the operator results.

Essential norms of composition operators and Aleksandrov measures

culated in terms of the Aleksandrov measures of the inducing holomorphic map. The argument provides a purely functiontheoretic proof of the equivalence of Sarason’s compactness condition for composition operators on L 1 and Shapiro’s compactness condition for composition operators on Hardy spaces. An application is given relating the essential norm to angular derivatives. x1. If is a holomorphic map of the unit disk D into itself, it is a consequence of Littlewood’s subordination principle [5] that composition with induces a bounded operator C on each Hardy space H p . A recurring theme in the study of composition operators has been the search for function theoretic conditions on which guarantee the compactness of C on H p .I t was shown by Shapiro and Taylor [11] that if C is compact on H p for some 0 <p< 1, then C is compact on H p for all 0 <p< 1, and so it is enough to study compactness on H 2 . In this context Shapiro [9] gave an expression for the essential norm of C on H 2 in terms of the Nevanlinna counting function of , thus providing a complete function theoretic characterization of compact composition operators on H 2 .

Hankel operators on the Bergman space of the unit polydisc

Let D D be the unit polydisc in C n {\mathbb {C}^n} . Let H 2 ( D ) {H^2}(D) be the Bergman space of D D . In this paper, by using the integral representations of solutions to the ∂ ¯ \overline \partial -equations, we give function theoretic characterizations of functions f ∈ L 2 ( D ) f \in {L^2}(D) such that the Hankel operators H f {H_f} from the Bergman space to L 2 ( D ) {L^2}(D) are bounded and compact, respectively.