p-representable operators in Banach spaces (Q1819712)

Let E and F be Banach spaces. An operator \(T\in L(E,F)\) is called p- representable if there exists a finite meausre \(\mu\) on the unit ball, \(B(E^*)\), of \(E^*\) and a function \(g\in L^ q(\mu,F)\), \(\frac{1}{p}+\frac{1}{q}=1\), such that \[ Tx=\int_{B(E^*)}<x,x^*>g(x^*)d\mu (x^*) \] for all \(x\in E\). The object of this paper is to investigate the class of all p- representable operators. In particular, it is shown that p-representable operators form a Banach ideal which is stable under injective tensor product. A characterization via factorization through \(L^ p\)-spaces is given.