Spectral results for operators commuting with translations on Banach spaces of sequences on Zᴷ and Z⁺

Abstract

We study the spectrum of multipliers (bounded operators commuting with the shift operator S) on a Banach space E of sequences on Z. Given a multiplier M, we prove that Mf(σ(S)) ⊂ σ(M) where Mf is the symbol of M. We obtain a similar result for the spectrum of an operator commuting with the shift on a Banach space of sequences on Z⁺. We generalize the results for multipliers on Banach spaces of sequences on Z^k.