On mappings between Banach spaces preserving $\MATHCAL A$-compact sets

TURCO, PABLO

Valencia

Congreso; Conference on non linear functional analysis; 2017

In the recent years, the behavior of polynomials and analytic functions on certain classes of compact sets received the attention of several authors. Namely, Aron and Rueda~\cite{AR} showed that homogeneous polynomials map $p$-compact sets into $p$-compact sets. Later,Aron, \c{C}ali\c{s}kan, Garc\'{\i}a and Maestre~\cite{ACGM} addressed a similar question for holomorphic mappings.Inspired in these results we are interested in the following questions. Fixed a class of $\mathcal A$-compact sets (of Carl and Stephani~\cite{CS}) where $\mathcal A$ is a $\lambda$-Banach operator ideal: Does every homogeneous polynomial between Banach spaces preserve $\mathcal A$-compact sets? When this not the case, is there any subclass of homogeneous polynomials for which the answer is positive?In this talk, making use of the theory of \textit{tensorstability} for operator ideals, we give some conditions under which homogeneous polynomials preserve $\mathcal A$-compact sets. We also show several examples. Our approach allows us to extend the obtained results to holomorphic functions between Banach spaces.Joint work with Silvia Lassalle (Universidad de San Andr\'es and CONICET).