Little Grothendieck's theorem for sublinear operators

Abstract

Let SB(X, Y ) be the set of the bounded sublinear operators from a Banach space X into a Banach lattice Y . Consider π 2 (X, Y ) the set of 2-summing sublinear operators. We study in this paper a variation of Grothendieck's theorem in the sublinear operators case. We prove under some conditions that every operator in SB(C(K), H ) is in π 2 (C(K), H ) for any compact K and any Hilbert H . In the noncommutative case the problem is still open.

Spaces generated by the cone of sublinear operators

“…We will denote by ∇T the subdifferential of T, which is the set of all linear operators u : X −→ Y such that u(x) ≤ T(x) for all x in X. We know (see, for example, [1]), that ∇T is not empty if Y is a complete Banach lattice and T(x) = sup{u(x) : u ∈ ∇T}, moreover, the supremum is attained. If Y is simply a Banach lattice, then ∇T is empty in general (see [3]).…”

“…Let T be in SL(X, Y). We have (see [1]), that T is bounded if and only if u is bounded for all u in ∇T. The set SL(X, Y) (respectively the space L(X, Y)) is a subset (respectively a subspace) of the space H(X, Y) of all homogeneous bounded operators from X into Y.…”