A type of Volterra operator

Joseph A Cima

doi:10.1186/s40627-016-0007-9

Abstract

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 \(H^2(\mathbb {D}).\)

Highlights

Introduction and definitions We useD as the notation for the unit disk in the complex plane and T as its boundary
The classical Hardy space on D is written as H p(D) and consists of those analytic functions f for which sup |f (r exp(ıθ ))|pdμ(θ ) < ∞
When p = 2, this is Hilbert space and identifying the Taylor coefficients of such an f with a sequence we obtain an isometry from H 2(D) onto the classical sequence space l2

Summary

Introduction

Introduction and definitions We useD as the notation for the unit disk in the complex plane and T as its boundary. The classical Hardy space on D is written as H p(D) and consists of those analytic functions f for which sup |f (r exp(ıθ ))|pdμ(θ ) < ∞. 2 The main result In [2], the authors study the integral operator (Hf )(z) =

Objectives

Results

Conclusion