Invariant subspaces and unstarred operator algebras

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.The results to be obtained will now be described in greater detail.Theorem 1 refines the following well-known result.THEOREM 0. If A is a normal operator on a Hilbert space H, and if the operator B on H commutes with every projection that commutes with A, then B belongs to the weakly closed star-algebra generated by A and the identity.