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A Sufficient Condition that an Operator Algebra be Self-Adjoint

Published online by Cambridge University Press:  20 November 2018

Heydar Radjavi  and
Peter Rosenthal
Heydar Radjavi
Affiliation:
Pahlavi University, Shiraz, Iran
Peter Rosenthal
Affiliation:
University of Toronto, Toronto, Ontario
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It is well-known, and easily verified, that each of the following assertions implies the preceding ones.

  1. (i) Every operator has a non-trivial invariant subspace.

  2. (ii) Every commutative operator algebra has a non-trivial invariant subspace,

  3. (iii) Every operator other than a multiple of the identity has a non-trivial hyperinvariant subspace.

  4. (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 .

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 4 , August 1971 , pp. 588 - 597
Copyright
Copyright © Canadian Mathematical Society 1971

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