Operators of Fourier type p with respect to some subgroups of a locally compact Abelian group

Abstract

It is shown that all Banach space operators which have Fourier type p (1 < p < 2) with respect to a second countable locally compact abelian group G also have Fourier type p with respect to every closed discrete subgroup H of G. The same statement holds for any closed subgroup H of G when p = 2. Also shown as a corollary is that Fourier type 2 operators have Walsh type 2.