The splitting property for \((p,S)\)-summing operators
Let \(X,Y\) be Banach spaces. An absolutely \(\left( p;q\right) \)-summing operator is a continuous linear operator \(T:X\rightarrow Y\) such that \( \left( T(x_{j}\right) )_{j=1}^{\infty }\in \ell_{p}(Y)\) whenever \( \sup_{\varphi \in B_{Y^{\ast }}}\sum\limits_{j=1}^{\infty }\left| \varphi (x_{j})\right| ^{q}<\infty , \) where \(B_{Y^{\ast }}\) denotes the closed unit ball of the dual of \(Y.\) There are several possible extensions of this notion to the multilinear setting: absolutely summing multilinear operators, multiple summing multilinear operators, dominated multilinear operators, strongly summing multilinear operators, strongly multiple summing multilinear operators, etc. In this paper, the author introduces the interesting notion of \((p,S)\)-summing operators. Several properties are proved, such as a domination theorem, a splitting lemma and a composition theorem.
scientific article; zbMATH DE number 6679195
scientific article