Triangular form for bounded linear operators
The canonical Jordan form for matrices has been extended in many ways to bounded operators on infinite dimensional Hilbert spaces. The spectral operators T of N. Dunford constitute a generic example from that point of view: \(T=S+Q\), where S is ``diagonalizable'' by a spectral measure, Q is quasi-nilpotent and \([S,Q]=0\), see \textit{N. Dunford}, and \textit{J. T. Schwartz}, Linear operators. Part III, New York (1971; Zbl 0243.47001). The aim of the paper under review is to prove the existence of a similar decomposition for operators T with sufficiently many invariant subspaces which divide the spectrum of T. Quite specifically, if there exists a finitely additive spectral measure of T with respect to a Boolean algebra of subsets of \({\mathbb{C}}\) which contains arbitrary small sets (in the sense of the diameter), then \(T=N+Q\), where N is a normal operator and Q is quasi-nilpotent. Moreover, it is shown that \(\sigma (N)=\sigma (T)\) and there exists a nest of invariant subspaces of N which separates the points of \(\sigma\) (T). The approach is similar but not reducible to the classical theory of Volterra operators, in the sense of Gohberg-Krein and Brodskij. A similar decomposition is proved to hold for Riesz elements in \(C^*\)-algebras modulo closed ideals. The paper ends with two interesting open problems.
bounded operators on infinite dimensional Hilbert spaces
Riesz elements in \(C^ *\)-algebras modulo closed ideals
scientific article; zbMATH DE number 4165782
scientific article