Fredholmness and index of operators in the Wiener algebra are independent of the underlying space

Abstract

The purpose of this paper is to demonstrate the so-called Fredholm-inverse closedness of the Wiener algebra W and to deduce independence of the Fredholm property and index of the underlying space. More precisely, we look at operators A ∈ W as acting on a family of vector valued p spaces and show that the Fredholm regularizer of A for one of these spaces can always be chosen in W as well and therefore regularizes A (modulo compact operators) on all of the p spaces under consideration. We conclude that both Fredholmness and the index of A do not depend on the p space that A is considered as acting on.