Extensions of Grothendieck and Bennett–Carl theorems

Abstract

Grothendieck has proved that the inclusion operator \(J:l_{1}\hookrightarrow l_{2}\) is 1-summing. Bennett and Carl proved, independently, that if \(1\le p\le q\le 2\) and \(\frac{1}{s}=\frac{1}{p}-\frac{1}{q}+\frac{1}{2}\), then the inclusion operator \(J:l_{p}\hookrightarrow l_{q}\) is (s, 1) -summing. In this paper we prove the following extension of Grothendieck theorem: If X, Y are Banach spaces and \(V:X\rightarrow Y\) is 1-summing then, the multiplication operator \(M_{V}:l_{1}[X] \rightarrow l_{2} (Y)\) is 1 -summing. Using this result we prove the following extension of Bennett and Carl theorem: If X and Y are Banach spaces and \(V:X\rightarrow Y\) is 1-summing then, the multiplication operator \(M_{V}:l_{p}(X) \rightarrow l_{q} (Y)\) is (s, 1)-summing. \(l_{1}[X]\), \(l_{p}(X)\) denotes the Banach spaces of all unconditionally norm convergent series respectively p-absolutely convergent series and \(M_{V}(( x_{n})_{n\in \mathbb {N}}) := (V(x_{n}))_{n\in \mathbb {N}}\) is the multiplication operator.