Behavior of holomorphic mappings on \(p\)-compact sets in a Banach space (Q2790640)

Given a Banach space \(X\) and \(1\leq p<\infty\) the authors use \(\ell_p(X)\) to denote the space of sequences \((x_n)_n\) in \(X\) such that \(\sum_{n=1}^\infty\|x_n\|^ p<\infty\). They say that a subset \(K\) of \(X\) is relatively \(p\)-compact if there is a sequence \((x_n)_n\) in \(\ell_p(X)\) with \(K\subset p\text{-co}(x_n)_n:=\left\{ \sum_{n=1}^\infty a_n x_n:(a_n)_n\in \overline{B}_{\ell_{p'}}\right\}\) (\({1\over p}+ {1\over p'}=1\)) and \(p\)-compact if it is closed and relatively \(p\)-compact. The main result of the paper is that entire holomorphic mappings between Banach spaces map \(p\)-compact sets to \(p\)-compact sets. An example is given to show that this result does not extend to continuous functions. It is also shown that if \(K\) is a compact subset of the \(m\)-fold projective tensor product, \(\widehat{\bigotimes}_{m,s,\pi}X\), then there is a sequence \((y_k)_k\) in \(\ell_{pm}(X)\) such that \(K\subset p\text{-co}(y_k\otimes\cdots\otimes y_k)_k\). A continuous linear operator \(T: X\to Y\) is said to be \(p\)-compact if there is a sequence \((y_n)_n\) in \(Y\) such that \(T(\overline{B}_X)\subseteq p\text{-co} (y_n)_n\). The infimum of the \(\ell_p\) norm of all such \((y_n)_n\) provides a norm for the space of \(p\)-compact operators from \(X\) to \(Y\). For each \(R>0\), an example is given of a linear operator between \(\ell_p\) spaces with operator norm at most \(1\) but which has \(p\)-compact norm greater than \(R\).