Vol. 17, No. 3, 1966

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Invariant subspaces and unstarred operator algebras

Donald Erik Sarason

Vol. 17 (1966), No. 3, 511–517

It is proved in the present paper that if A is a normal Hilbert 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. This may be regarded as a refinement of the von Neumann double commutant theorem. A generalization is given in which the single operator A is replaced by a commuting family of normal operators. Also the same result is proved for the case where A is an analytic Toeplitz operator.

Mathematical Subject Classification
Primary: 47.35
Secondary: 46.65
Received: 7 January 1965
Published: 1 June 1966
Donald Erik Sarason