Journal ArticleDOI
On Reductive Operators and Operator Algebras
A I Loginov,V S Šul'man +1 more
TLDR
In this article, a theorem on the structure of weakly closed reductive operator algebras was proved, based on a known result of V. I. Lomonosov on transitive operators containing a nonzero compact operator.Abstract:
We prove a theorem on the structure of weakly closed reductive operator algebras. The proof essentially relies on a known result of V. I. Lomonosov on transitive operator algebras containing a nonzero compact operator. We deduce a number of corollaries which apply to the reductivity problem.Bibliography: 20 titles.read more
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Completely reducible operators that commute with compact operators
TL;DR: Azoff and Lomonosov as mentioned in this paper showed that every completely reducible operator commuting with an injective compact operator with a dense range is a scalar type spectral operator, and that the weakly closed unital algebra generated by such an operator must be reflexive.
Journal ArticleDOI
Invariant subspaces of operator algebras
A. I. Loginov,V. S. Shul'man +1 more
TL;DR: In this paper, a survey is devoted to a circle of problems, grouped around one of the oldest problems of functional analysis, namely the invariant subspace problem, and it is shown that the investigation of the algebraic and analytic properties of families of operators touches upon the question of the structure of their invariants.
That commute with compact operators
TL;DR: In this article, it was shown that every completely reducible operator commuting with an injective compact operator with a dense range is a scalar type spectral operator and that the weakly closed unital algebra generated by such an operator must be reflexive.
References
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Journal ArticleDOI
Invariant subspaces and unstarred operator algebras.
TL;DR: In this article, it was shown that if A is a normal Hubert space operator, and if the operator B leaves invariant every invariant subspace of A, then B belongs to the weakly closed algebra generated by A and the identity.
Journal ArticleDOI
Invariant subspaces for the family of operators which commute with a completely continuous operator
Journal ArticleDOI
A sufficient condition that an operator algebra be self-adjoint
Heydar Radjavi,Peter Rosenthal +1 more
TL;DR: In this article, it was shown that every operator has a non-trivial invariant subspace, and every operator other than a multiple of the identity has a hyper-invariant hyper-subspace.
Journal ArticleDOI
An equivalent formulation of the invariant subspace conjecture
TL;DR: The invariant subspace conjecture for bounded linear operators acting on a Hubert space has been studied in this article, where it has been shown that such operators have the property of having a reducing subspace.