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

TURCO, PABLO; LASSALLE, SILVIA

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Año: 2018 vol. 463 p. 1092 - 1108

0022-247X

In this paper we study the behavior of holomorphic mappings on $A$-compact sets. Motivated by the recent work of Aron, c{C}alic{s}kan, Garc´{i}a and Maestre (2016), we give several conditions (on the holomorphic mappings and on the $lambda$-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.