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Journal of Operator Theory

Volume 63, Issue 2, Spring 2010  pp. 283-300.

Multipliers and Toeplitz operators on Banach spaces of sequences

Authors Violeta Petkova
Author institution: LABAG, Univ. Bordeaux I, 351, Cours de la Liberation, 33405 Talence, France; Actual address: LMAM, Universite de Metz, Ile du Saulcy, 57045, Metz, France

Summary:  In this paper we prove that every multiplier $M$ (every bounded operator commuting with the shift operator $S$) on a large class of Banach spaces of sequences on $\Z$ is associated to a function essentially bounded by $\|M\|$ on $\mathrm{spec}(S)$. This function is holomorphic on $\overset{\circ}{\mathrm{spec}}(S)$ if $\overset{\circ}{\mathrm{spec}}(S)\neq \emptyset$. Moreover, we give a simple description of $\mathrm{spec}(S)$. We also obtain similar results for Toeplitz operators on a large class of Banach spaces of sequences on~$\Z^+$.

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