Operator ideals and spaces of bilinear operators

Multilinear extensions of Grothendieck's theorem

“…In this note we introduce a new definition of p-summing multilinear operators (although the origin of this definition can be found in [17,Definition 3.6]), which we call multiple p-summing operators. In this paper we focus on Grothendieck type theorems related to this class.…”

“…Later on, motivated by the growth of the multilinear and polynomical theory of Banach spaces, different authors started the study of the p-summing multilinear operators between Banach spaces (see [1], [6], [10], [15], [17]). For some of these psumming multilinear operators a factorization result extending Pietsch DominationFactorization Theorem [5, Theorems 2.12 and 2.13] holds, but no multilinear version of Grothendieck's Theorem in terms of these operators is known, except for a special case presented in [2] (see also [11]).…”

Absolutely continuous multilinear operators

“…As a consequence of the factorization properties of compact bilinear maps that can be found in [22] and [23], we obtain the following These operators are characterized as the ones that allow a domination by means of an interpolation formula as follows (see Theorem 3.4 in [27]). Suppose that X(µ) is order continuous.…”

Quasi s-numbers and measures of non-compactness of multilinear operators

“…The main results of this section states that, up to universal constants, the measure of non-compactness of every multilinear operator T is equivalent to the measure of non-compactness of its generalized adjoint (adjoint for short) operator T × of T . As a consequence we obtain a variant of Schauder's theorem for multilinear operators, which was first proved for the bilinear case by Ramanujan and Schock [15].…”

“…A slightly more general result says that the Kuratowski-Hausdorff measure of non-compactness of an operator T is equivalent to the measure of non-compactness of its adjoint. Ramanujan and Schock studied in [15] ideals of bilinear operators between Banach spaces, including the ideal of bilinear compact operators, …”