Geodesic
A Sufficient Condition that an Operator Algebra be Self-Adjoint
Canadian journal of mathematics, Tome 23 (1971) no. 4, pp. 588-597

Voir la notice de l'article provenant de la source Cambridge University Press

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 . 
Radjavi, Heydar; Rosenthal, Peter. A Sufficient Condition that an Operator Algebra be Self-Adjoint. Canadian journal of mathematics, Tome 23 (1971) no. 4, pp. 588-597. doi: 10.4153/CJM-1971-066-7
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