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On the transformation of vector-valued sequences by linear and multilinear operators. (English) Zbl 1434.46026
The authors of the paper under review provide a unified approach to the study of Banach ideals of linear and multilinear operators defined, or characterized, by transformations of vector-valued sequences. Their abstract approach is (besides smart and interesting) quite complete and broad since, for instance, it covers, among others, the following classes: \(p\)-dominated \(n\)-linear operators, absolutely \((s; r )\)-summing linear and multilinear operators, unconditionally \(p\)-summing linear and multilinear operators, almost summing linear and multilinear operators, weakly sequentially continuous multilinear operators, or Cohen strongly \(p\)-summing multilinear operators. The authors also investigate linear and multilinear stabilities of some frequently used classes of vector-valued sequences, and provide particular useful and clarifying applications of their work.

MSC:
46G25 (Spaces of) multilinear mappings, polynomials
47L22 Ideals of polynomials and of multilinear mappings in operator theory
46B45 Banach sequence spaces
46B25 Classical Banach spaces in the general theory
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