Invariant subspaces and unstarred operator algebras

Abstract

It is proved in the present paper that if A is a normal Hubert 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.