A Sufficient Condition that an Operator Algebra be Self-Adjoint

Abstract

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 .