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# A type of Volterra operator

- Joseph A. Cima
^{1}Email author

**2**:3

https://doi.org/10.1186/s40627-016-0007-9

© The Author(s) 2016

**Received: **27 May 2016

**Accepted: **7 September 2016

**Published: **16 September 2016

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

## Keywords

- Hardy spaces
- Volterra type operators

## Mathematics Subject Classification

- 30J10
- 47A66

## 1 Introduction and definitions

## 2 The main result

The purpose of this paper is to discuss the range of \(\mathcal {H}\) on \(H^1\). The main result is the following.

###
**Theorem 1**

*The operator *
\(\mathcal {H}\)
* maps*
\(H^1\)
* to the space of Cauchy transforms of measures on the unit circle*
\(\partial \mathbb {D}\).* Furthermore,*
\(\mathcal {H}\)
* is injective*

*M*be the Banach space of finite complex Borel measures on \(\partial \mathbb {D}\) endowed with the total variation norm \(\Vert \mu \Vert \) and define

###
*Proof of Theorem 1*

*p*. The Weisrstrass approximation theorem and the fact that \(f \in L^1[0, 1]\) yields

*g*on [0, 1]. By the Riesz representation theorem for the space of finite Borel measures on [0, 1], we conclude that \(f(t) = 0\) almost everywhere on [0, 1]. The identity theorem for analytic functions implies that \(f \equiv 0\). \(\square \)

###
*Remark 1*

*q*is the Holder conjugate index for

*p*, via the pairing

###
*Remark 2*

## Declarations

### Acknowledgements

The author would like to thank the referee for a careful rewriting of the principal result of this paper.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Duren, P.: Theory of H
^{p}Spaces, Pure and Applied Mathematics**11**. Academic Press, New York (1970)Google Scholar - Diamantopoulos, E., Siskakis, A.G.: Composition operators and the Hilbert matrix. Studia Math
**140**, 191–198 (2000)MathSciNetMATHGoogle Scholar - Cima, J.A., Matheson, A.L., Ross, W.T.: The Cauchy Transform, Mathematical Surveys and Monographs,
**125**. American Mathematical Society, Providence (2006)View ArticleGoogle Scholar