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Archiv der Mathematik

, Volume 112, Issue 1, pp 71–82 | Cite as

Convolution operators in discrete Cesàro spaces

  • Werner J. RickerEmail author
Article
  • 34 Downloads

Abstract

The discrete Cesàro spaces \({{\text {ces(p )}}}, 1< p < \infty ,\) are well known and arise from the classical sequence spaces \(\ell ^p,1<p<\infty \), via the process of averaging. It is known that a sequence \(b\in {\mathbb {C}}^{\mathbb {N}}\) convolves \({{\text {ces(p )}}}\) into itself if and only if \(b\in \ell ^1\), which is a very different situation than for convolution in the spaces \(\ell ^p\). The purpose of this note is to determine the spectrum of the convolution operator \(a \mapsto b*a\), for \(a\in {{\text {ces(p )}}}\), whenever \(b\in \ell ^1\). It is then possible to describe the mean ergodic properties of such a convolution operator.

Keywords

Convolution operator Spectrum Discrete Cesàro space Banach algebra Mean ergodicity 

Mathematics Subject Classification

Primary 42A85 43A22 47A10 47B37 Secondary 47A35 47C05 

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Notes

Acknowledgements

Thanks go to the referee for some useful comments and suggestions.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Math.-Geogr. FakultätKatholische Univ. Eichstätt-IngolstadtEichstättGermany

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