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Melnichenko, G.

Structure of certain operators from a nest algebra. (English. Russian original) Zbl 0573.47040

Lith. Math. J. 22, 408-416 (1983); translation from Litov. Mat. Sb. 22, No. 4, 98-108 (1982).
Let \({\mathcal L}(H)\) denote the algebra of all bounded linear operators on a complex Hilbert space H and let \(A\in {\mathcal L}(H)\) be similar to a self- . Then A gives rise in a natural way to a complete chain \({\mathcal G}\) of on H and A belongs to the corresponding \({\mathcal N}\). The main aim of the present note is to obtain the decomposition \(A=A_ 0+A_+\), where the \(A_ 0\) belongs to the von-Neumann algebra generated by \({\mathcal G}\) and \(A_+\) lies in the radical of \({\mathcal N}\). A corresponding multiplicative decomposition for an operator similar to a unitary one is also given. The paper ends with a proof of the existence of a of \({\mathcal L}(H)\) onto the diagonal of an arbitrary nest algebra on H.
Reviewer: T.A.Gillespie

MSC:

47L30 Abstract operator algebras on Hilbert spaces
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
47A15 Invariant subspaces of linear operators
47C15 Linear operators in \(C^*\)- or von Neumann algebras

Keywords:

similar to a self-adjoint operator; nest algebra; von-Neumann algebra; radical; multiplicative decomposition; conditional expectation
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References:

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