A general version of the Saphar theorem for linear summing operators

Source abstract

Abstract Let $$1\leqslant p<\infty$$ 1 ⩽ p &lt; ∞ , E , W be Banach spaces, $$\Omega$$ Ω a non-empty set, $$F\subseteqq l_{\infty }( \Omega ,W)$$ F ⫅ l ∞ ( Ω , W ) a linear subspace, $$S:E\rightarrow F$$ S : E → F a positive homogeneous operator and Z a Banach space. In this paper we introduce the concept of ( p , S ) -summing operator from E into Z and on $$E\hspace{1.111pt}{\otimes }\hspace{1.111pt}Z$$ E ⊗ Z we define $$d_{p^{*}}^{S}$$ d p ∗ S , the general droite Saphar seminorm associated to the operator S . We prove that the spaces $$\Pi _{p}^{S}( E,Z^{*})$$ Π p S ( E , Z ∗ ) and $$( E\hspace{1.111pt}{\otimes }\hspace{1.111pt}Z,d_{p^{*}}^{S}) ^{*}$$ ( E ⊗ Z , d p ∗ S ) ∗ are isometrically isomorphic. Various applications and illustrative examples are given.