Cohomology invariants of collections of essentially commuting operators
If T is an operator n-tuple in a Banach space X, in [Spectral theory, Warsaw 1977, Banach Cent. Publ. 8, 321-332 (1982; Zbl 0496.47017)], the author introduced and investigated Taylor's joint essential spectrum \(\sigma_ e(T)\) of T in terms of sheaf theory [for the usual definition, see \textit{J. L. Taylor,} J. Funct. Anal. 6, 172-191 (1970; Zbl 0233.47024)]. In this paper to each n-tuple T whose commutators are compact, corresponds an element of the K-group of the complement \(C^ n\backslash \sigma_ e(T)\). If T is commuting, it is proved that this element is generated by a coherent sheaf. No proofs are provided. The obtained results imply an \textit{L. Boutet de Monvel} formula on the index of a Toeplitz operator [see Invent. Math. 50, 249-272 (1979; Zbl 0398.47018)].
cohomology invariants of collections of essentially commuting operators
scientific article; zbMATH DE number 3846248