Description of \(\operatorname {AlgLat}T/W(T)\) for certain extensions of selfadjoint operators
type: Item
instance of: scholarly article
title: Description of \(\operatorname {AlgLat}T/W(T)\) for certain extensions of selfadjoint operators
zbMATH Open document ID: 0984.47050
DOI: 10.1016/S0024-3795(00)00210-X
author: M'hammed Benlarbi Delai
published in: Linear Algebra and its Applications
publication date: 2002-04-21T00:00:00Z
review text: Let \(H\) and \(K\) be Hilbert spaces. The operator \(T\) mentioned in the title is of the form \(T=\left(\begin{smallmatrix} A&X\\ 0&N\end{smallmatrix}\right)\), where \(A\) is a selfadjoint operator in \({\mathcal L}(H)\) such that \(0\) is not in the point spectrum of \(A\), \(N\in {\mathcal L}(K)\) is a cyclic nilpotent operator of order \(n\), and \(X\) belongs to \({\mathcal L}(K, H)\). The symbol \(W(T)\) in the title means the weak closure of the polynomials in \(T\). Let \(e\in K\) be a cyclic vector for \(N\). The description mentioned in the title is obtained under the condition that the minimal closed subspace invariand under \(A\) and containing \(T^n e\) is of finite codimension in \(H\). As a consequence it is shown that, under the above conditions, the quotient mentioned in the title is of dimension at most \(n(n-1)/2\), and a necessary and sufficient is given for this dimension to be \(0\).
Mathematics Subject Classification ID: 47L30
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 47L35
zbMATH DE Number: 1559340
zbMATH Keywords: operator algebra
zbMATH Keywords: lattice of invariant subspaces
zbMATH Keywords: cyclic nilpotent operator
reviewed by: S. V. Kislyakov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1559340