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

It is shown that all Banach space operators which have Fouriertype p(1 < p 2) with respect to a second countablelocally compact abelian group G also have Fourier type p withrespect to every closed discrete subgroup Hof G. The samestatement holds for any closed subgroup Hof G when p = 2.Also shown as a corollary is that Fourier type 2 operators haveWalsh type 2.Received: 10 June 2002