Convolution Operators on Banach Lattices with Shift-Invariant Norms

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][106][107][108][109][110][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.Mathematics Subject Classification (2010). Primary 47A30, 47B38; Secondary 43A15.

Cross-Toeplitz Operators on the Fock--Segal--Bargmann Spaces and Two-Sided Convolutions on the Heisenberg Group

“…Of course, L p -norms, 1 p ∞, are invariant under spatial and phase shift. There are other invariant norms, which are yet less studied, for example [44,70]:…”