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Tensor products and elementary operators. (English) Zbl 0639.47003

An axiomatic tensor product notion is developed which applies at the same time to operator ideals between Banach spaces and to ordinary tensor products of Banach spaces formed with respect to a quasi-uniform crossnorm. For this tensor product a Künneth formula is obtained which relates the homology group of a total differential \(d=d_ 1\otimes I+\eta \otimes d_ 2\) to the homology groups of the single differentials \(d_ 1\), \(d_ 2.\)
As an application the Taylor spectra and essential Taylor spectra of tensor product tuples of the form (S\(\otimes I, I\otimes T)\) are described, where \(S\in L(X)^ n\), \(T\in L(Y)^ m\) are tuples of commuting Banach space operators. Spectral mapping theorems yield, for example, the spectra and essential spectra of elementary operators acting on operator ideals between Banach spaces. For single operators \(S\in L(X)\), \(T\in L(Y)\) an analytic index formula is developed which allows to compute the index of operators of the form f(S\(\otimes I, I\otimes T)\), \(f\in {\mathcal O}(\sigma (S\otimes I, I\otimes T))\) in its non-essential spectral points.
Reviewer: J.Eschmeier

MSC:

47A10 Spectrum, resolvent
47B47 Commutators, derivations, elementary operators, etc.
46M05 Tensor products in functional analysis
47A53 (Semi-) Fredholm operators; index theories
47L10 Algebras of operators on Banach spaces and other topological linear spaces
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