Cyclic vectors and invariant subspaces for unbounded multiplication operators on Hilbert space

Abstract

Sufficient conditions are given for a vector to be cyclic, or more generally, to generate a spectral subspace for a densely defined unbounded multiplication operator acting on a Hilbert space having an orthonormal basis of eigenvectors with unbounded associated set of eigenvalues. Such operators are shown to have non-spectral invariant subspaces which are closed with respect to a topology defined on the core of the operator. Conditions are also given for such operators to have non-spectral norm-closed invariant subspaces.