On a generalization of the Jordan canonical form theorem on separable Hilbert spaces

Abstract

We prove a generalization of the Jordan canonical form theorem for a class of bounded linear operators on complex separable Hilbert spaces.2000 Mathematics Subject Classification. Primary 47A65, 47A67; Secondary 47A15, 47C15.

A reduction theory for operators in type $\rm{I}_{n}$ von Neumann algebras

“…The multiplicity function for N plays an important role, and it is required to be bounded if we want bounded maximal abelian sets of idempotents in {N } ′ to be the same up to similarity. Our results in [10,7] are proved based on the 'bounded multiplicity' condition. On the other hand, another way is to consider the generalization in type I n von Neumann algebras.…”

“…One way is to consider the generalization in the type I ∞ factor instead of the type I n factor M n (C). We started our study in [6] and carried on in [10,7]. In this 'infinite' case, we found that there exists a normal operator N such that there are two bounded maximal abelian sets of idempotents P and Q in {N } ′ not similar to each other in {N } ′ .…”

A note on curvature and similarity of some Cowen–Douglas operators

“…These results are also generalized to the case of direct integrals of strongly irreducible operators by R. Shi (cf. [7]). …”

A similarity invariant of a class of n-normal operators in terms of K-theory

“…In this paper the authors continue the study on generalizing the Jordan canonical form theorem for bounded linear operators on separable Hilbert spaces, which was initiated in [9] and carried on in [12]. Throughout this article, we only discuss Hilbert spaces which are complex and separable.…”

“…Inspired by (1.1), our second question is whether the equality (1.1) holds in the commutant {A} ′ for each operator A in S . In [12], we gave a negative answer by constructing an operator C in the form…”