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A Sufficient Condition that an Operator Algebra be Self-Adjoint
Published online by Cambridge University Press: 20 November 2018
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It is well-known, and easily verified, that each of the following assertions implies the preceding ones.
(i) Every operator has a non-trivial invariant subspace.
(ii) Every commutative operator algebra has a non-trivial invariant subspace,
(iii) Every operator other than a multiple of the identity has a non-trivial hyperinvariant subspace.
(iv) The only transitive operator algebra on is
Note. Operator means bounded linear operator on a complex Hilbert space , operator algebra means weakly closed algebra of operators containing the identity, subspace means closed linear manifold, a non-trivial subspace is a subspace other than {0} and , a. hyperinvariant subspace for A is a subspace invariant under every operator which commutes with A, a transitive operator algebra is one without any non-trivial invariant subspaces and denotes the algebra of all operators on .
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- Copyright © Canadian Mathematical Society 1971
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