Multipliers on homogeneous Banach spaces with respect to Jacobi polynomials

Abstract

A bounded linear operator is called multiplier with respect to Jacobi polynomials if and only if it commutes with all Jacobi translation operators on \([-1,1]\). Multipliers on homogeneous Banach spaces on \([-1,1]\) determined by the Jacobi translation operator are introduced and studied. First we prove four equivalent characterizations of a multiplier for an arbitrary homogeneous Banach spaces \(B\) on \([-1,1]\). One of them implies the existence of an algebra isomorphism from the set of all multipliers on \(B\) into the set of all pseudomeasures. Further, we study multipliers on specific examples of homogeneous Banach spaces on \([-1,1]\). Amongst others, multipliers on the Wiener algebra, on the Beurling space and on Sobolev spaces are analyzed. We obtain that the multiplier spaces of the Wiener algebra, the Beurling space and of all Sobolev spaces are isometric isomorphic to each other. Furthermore, these multiplier spaces are all isometric isomorphic to the set of all pseudomeasures.