Little Grothendieck's theorem for sublinear operators
A sublinear operator \(T:X\rightarrow Y\) between a Banach space \(X\) and a Banach lattice \(Y\) is called \(2\)-summing, if the norms of the mappings \(\text{id} \otimes T: \ell_2^n \otimes_\varepsilon X \rightarrow \ell_2^n(Y)\) are uniformly bounded. Little Grothendieck's theorem states that every bounded linear operator from \(C(K)\) into a Hilbert space \(H\) is \(2\)-summing. In this article, the authors present some generalizations thereof within the framework of sublinear operators. In particular, they show that the theorem remains true for a bounded sublinear operator \(T:C(K) \rightarrow H\) if and only if it is \(2\)-regular, that is, \(\| (\sum_1^n | Tx_i| ^2)^{1/2}\| \leq C\), \(\| (\sum_1^n | x_i| ^2)^{1/2}\| \) for some \(C>0\) and all \(x_1, \ldots, x_n \in C(K)\). It is left open whether the assumption of \(2\)-regularity is necessary -- note that by a result of Krivine, every bounded linear operator between Banach lattices is \(2\)-regular.
sublinear operator, \(p\)-summing operator
scientific article; zbMATH DE number 2092944
scientific article