Space Of Cauchy Research Articles

Norm of the general polynomial differentiation composition operator from the space of Cauchy transforms to the mth weighted‐type space on the unit disk

We introduce the general polynomial differentiation composition operator where , , , are holomorphic functions on the open unit disk and , , are holomorphic self‐maps of , calculate norm of the operator acting from the space of Cauchy transforms to the th weighted‐type space, and characterize its boundedness, as well as the boundedness of the operator acting from the space of Cauchy transforms to the little th weighted‐type space.

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.

The Norm of a Composition Operator from the Space of Cauchy Transforms into Zygmund-Type Spaces

The problem of evaluation of the norm of a composition operator acting between the spaces of holomorphic functions is quite difficult. In the Hardy and weighted Bergman spaces, the norm of the composition operator is unknown even for the choice of a fairly simple symbol. We compute the operator norm of the composition operator acting between the space of Cauchy transforms and Zygmund-type spaces. We also characterize bounded and compact composition operators acting between these spaces.

Difference of Differentiation Composition Operators on the Fractional Cauchy Transforms Spaces

In this paper, we consider the boundedness and compactness of the differences of differentiation composition operators from the space of fractional Cauchy transforms to the Bloch-type spaces and the weighted Dirichlet spaces. Surprisingly, these characterizations are free from pseudo-hyperbolic metric, which is a common feature of all the preceding characterizations of difference of differentiation composition operators on various spaces of holomorphic functions.

Difference of generalized integration operators on the space of Cauchy transforms

ABSTRACTIn this paper, the boundedness and compactness of the differences of generalized integration operators from the space of Cauchy integral transforms to the Bloch-type spaces and the weighted Dirichlet spaces are investigated.

A type of Volterra operator

In Diamantopoulos and Siskakis (Studia Math 140:191–198, 2000), the authors study the action of the classical Cesaro matrix C on the Taylor coefficients of analytic functions on the Hardy spaces \(H^p(\mathbb {D}), \;\; 1<p < \infty .\) They convert the matricial action of C on sequences into a Volterra type integral operator \(\mathbb {H}\) on \(H^P.\) They show that it is bounded for \(1<p<\infty \) and derive estimates on the operator norm of \(\mathbb {H}.\) We continue this study and show that \(\mathbb {H}\) maps boundedly from \(H^1(\mathbb {D})\) into the space of Cauchy transforms of finite Borel measures on unit circle. We show that \(\mathbb {H}\) is one to one on \(H^2(\mathbb {D}).\)

Composition operators from the space of Cauchy transforms to Bloch and the little Bloch-type spaces on the unit disk

We completely characterize the boundedness and compactness of composition operators from the space of Cauchy transforms on the unit disk D , into the Bloch-type space B ν as well as the little Bloch-type space B ν , 0 , consisting respectively of all holomorphic functions f on D such that sup z ∈ D ν ( z ) | f ′ ( z ) | < ∞ , that is, of all holomorphic functions f on D such that lim | z | → 1 ν ( z ) | f ′ ( z ) | = 0 , for some weight function ν . As a byproduct of our results, norm of the operator is calculated when B ν is replaced by B ν / C .

An Observation about Frostman Shifts

A classical theorem of Frostman says that if B is a Blaschke product (or any inner function), then its Frostman shifts \(B_w = \left( {B - w} \right)\left( {1 - \bar wB} \right)^{ - 1}\) are Blaschke products for all |w| < 1 except possibly for w in a set of logarithmic capacity zero. If B is a Frostman Blaschke product, equivalently an inner multiplier for the space of Cauchy transforms of measures on the unit circle, we show that for all |w| < 1, Bw is indeed another Frostman Blaschke product.

The backward shift on the space of Cauchy transforms

This note examines the subspaces of the space of Cauchy transforms of measures on the unit circle that are invariant under the backward shift operator f → z − 1 ( f − f ( 0 ) ) f \to z^{-1}(f - f(0)) . We examine this question when the space of Cauchy transforms is endowed with both the norm and weak ∗ {}^* topologies.

Composition Operators from the Space of Cauchy Transforms into Its Hardy-Type Subspaces

Composition Operators from the Space of Cauchy Transforms into Its Hardy-Type Subspaces