Coincidence of extendible vector-valued ideals with their minimal kernel

Abstract

We provide coincidence results for vector-valued ideals of multilinear operators. More precisely, if A is an ideal of n-linear mappings we give conditions for which the following equality A(E 1 , . . . , E n ; F ) = A min (E 1 , . . . , E n ; F ) holds isometrically. As an application, we obtain in many cases that the monomials form a Schauder basis of the space A(E 1 , . . . , E n ; F ). Several structural and geometric properties are also derived using this equality. We apply our results to the particular case where A is the classical ideal of extendible or Pietsch-integral multilinear operators. Similar statements are given for ideals of vector-valued homogeneous polynomials.2010 Mathematics Subject Classification. 46G25,46B22, 46M05, 47H60.

On the transformation of vector-valued sequences by linear and multilinear operators

“…The notion of operator ideals was systematized by Pietsch [30] and multi-ideals were introduced by Pietsch [31] and has been developed by several authors (for recent contributions, see, e.g., [14,22,27,28,32,35]). Definition 2.5.…”

Type and cotype of multilinear operators

“…The case n = 1 recovers the theory of Banach/quasi-Banach operator ideals, for which we refer to [11,21]. For the multilinear case see, e.g., [5,6,7,15,16].…”

“…Acknowledgment. The authors thank D. Galicer for his helpful suggestions and for drawing our attention to reference [16].…”

Polynomial ideals from a nonlinear viewpoint

“…For the last three decades, a huge amount of work has been done on polynomial ideals. The theory has been developed pari passu with the theory of ideals of multilinear operators (multi-ideals), and connections have been established with other topics, such as (just a few references are given): infinite dimensional holomorphy [8,21,43], topological tensor products [23,30,42], ultrapower stability [40,41], quantum information theory [48,53], Dirichlet series [32,34], integral formulas/stable measures [22], coherence/compatibility [20,50], Bishop-Phelps-Bollobás-Lindenstrauss circle of ideas [2,24], unconditional bases [23,33], interpolation theory [4,18,35], classical inequalities (Hardy-Littlewood, Bohnenblust-Hille, Blei) [3,4,5,35,59], summability properties [1,47,51], approximation properties [8,29,30], extension of multilinear operators and polynomials [27,42,45], Aron-Berner stability [12,42], hypercyclic convolution operators [9,19], homological methods [27], lineability/spaceability [13,…”