Eschmeier, Jörg Tensor products and elementary operators. (English) Zbl 0639.47003 J. Reine Angew. Math. 390, 47-66 (1988). 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 Cited in 2 ReviewsCited in 22 Documents 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 Keywords:axiomatic tensor product; operator ideals between Banach spaces; ordinary tensor products of Banach spaces; quasi-uniform crossnorm; Künneth formula; homology group of a total differential; essential Taylor spectra of tensor product tuples; tuples of commuting Banach space operators; Spectral mapping theorems; essential spectra of elementary operators acting on operator ideals between Banach spaces; analytic index formula; non-essential spectral points PDFBibTeX XMLCite \textit{J. Eschmeier}, J. Reine Angew. Math. 390, 47--66 (1988; Zbl 0639.47003) Full Text: DOI Crelle EuDML