Polynomials and holomorphic functions on A-compact sets in Banach spaces

Abstract

In this paper we study the behavior of holomorphic mappings on A-compact sets. Motivated by the recent work of Aron, Çalişkan, García and Maestre (2016), we give several conditions (on the holomorphic mappings and on the λ-Banach operator ideal A) under which A-compact sets are preserved. Appealing to the notion of tensor stability for operator ideals, we first address the question in the polynomial setting. Then, we define a radius of (A;B)-compactification that permits us to tackle the analytic case. Our approach, for instance, allows us to show that the image of any (p,r)-compact set under any holomorphic function (defined on any open set of a Banach space) is again (p,r)-compact.