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Generalized inverses and special type operator algebras. (English) Zbl 1266.47003

Summary: Let \(\mathcal H\) be a complex Hilbert space. In this work, we compute the generalized inverse of a finite rank operator on \(\mathcal H\) and give necessary and sufficient conditions such that the generalized inverse of the product of two rank-1 operators is the product of the generalized inverses of the corresponding operators in reverse order. We also consider the generalized inverse of products of special type operators. We examine when the generalized inverse of a rank-1 operator in a nest algebra belongs to the nest algebra and give necessary conditions for an operator in a nest algebra with a continuous nest so that its generalized inverse belongs to the nest algebra. Finally, we give equivalent conditions so that an operator with closed range factors with respect to a closed subalgebra of a von Neumann algebra of operators on \(\mathcal H\).

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A46 Chains (nests) of projections or of invariant subspaces, integrals along chains, etc.
47C15 Linear operators in \(C^*\)- or von Neumann algebras
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