Operators defined on projective and natural tensor products
This paper studies the behaviour of operators defined on the projective and natural tensor product of an \(\ell_ p\) space and an arbitrary Banach space \(X\). The main result is the following: If \(E\) and \(F\) are Banach spaces, \(E\) not containing \(\ell_ 1\), and all operators from \(E\) into \(\ell_ p\) and all operators from \(E\) into \(F\) are compact, then all operators from \(E\) into \(\ell_ p\widehat\otimes_ \pi X\) are compact. Two applications of the techniques involved in the proof of this result are considered: a study of the tensor stability (with respect to the projective and natural tensor product with an \(\ell_ p\)-space) of the scale of operator ideals formed by the \(p\)-converging operators for \(1\leq p\leq +\infty\), and a vector-valued version of Pitt's theorem.
operators defined on projective and natural tensor products
scale of operator ideals formed by the \(p\)- converging operators
scientific article; zbMATH DE number 474979