Convolution operators on Banach lattices with shift-invariant norms

Abstract

Let G be a locally compact abelian group and let μbe a complex valued regular Borel measure on G. In this paper we consider a generalisation of a class of Banach lattices introduced in [6]. We use Laplace transform methods to show that the norm of a convolution operator with symbol μon such a space is bounded below by the L_\infty 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 μis bounded above by the total variation of μ.