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Ideals of multilinear functionals (designs of a theory). (English) Zbl 0562.47037

Operator algebras, ideals, and their applications in theoretical physics, Proc. int. Conf., Leipzig 1983, Teubner-Texte Math. 67, 185-199 (1984).
[For the entire collection see Zbl 0535.00014.]
The author outlines a general program for generalizing the theory of operator ideals to the setting of multilinear operators. Essentially, it suffices to consider “ideals” of multilinear (\(m\)-)functionals. The basic definition and some examples of such “ideals” are given, e.g., nuclear, integral and absolutely summing \(m\)-functionals. Starting with (usual) operator ideals \({\mathfrak A}_ 1,\dots,{\mathfrak A}_ n\), some methods are given to construct multilinear ideals from them, e.g., by requiring that the \(i\)-th component maps belong to \({\mathfrak A}_ i\). There are several natural possibilities to define the “rank” of a multilinear functional which lead to different possibilities to define approximation numbers or other \(s\)-numbers of \(m\)-functionals. At this point, not even a Hilbert space theory has yet been developed.
Reviewer: H.König

MSC:

47L10 Algebras of operators on Banach spaces and other topological linear spaces
47L30 Abstract operator algebras on Hilbert spaces

Citations:

Zbl 0535.00014