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Convolution Operators on Banach Lattices with Shift-Invariant Norms

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Abstract

Let G be a locally compact abelian group and let μ be a complex valued regular Borel measure on G. In this paper we consider a generalisation of a class of Banach lattices introduced in Johansson (Syst Control Lett 57:105–111, 2008). We use Laplace transform methods to show that the norm of a convolution operator with symbol μ on such a space is bounded below by the L ∞ norm of the Fourier–Stieltjes transform of μ. We also show that for any Banach lattice of locally integrable functions on G with a shift-invariant norm, the norm of a convolution operator with symbol μ is bounded above by the total variation of μ.

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    Nazar Miheisi

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  1. Nazar Miheisi
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Correspondence to Nazar Miheisi.

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Miheisi, N. Convolution Operators on Banach Lattices with Shift-Invariant Norms. Integr. Equ. Oper. Theory 68, 287–299 (2010). https://doi.org/10.1007/s00020-010-1817-4

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  • DOI: https://doi.org/10.1007/s00020-010-1817-4

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