The singularly continuous spectrum and non-closed invariant subspaces

Let A be a bounded self-adjoint operator on a separable Hilbert space h and h(0) subset of h a closed invariant subspace of A. Assuming that h(0) is of codimension 1, we study the variation of the invariant subspace h(0) under bounded self-adjoint perturbations V of A that are off-diagonal with respect to the decomposition h = h(0) circle plus h(1). In particular, we prove the existence of a one-parameter family of dense non-closed invariant subspaces of the operator A + V provided that this operator has a nonempty singularly continuous spectrum. We show that such subspaces are related to non-closable densely defined solutions of the operator Riccati equation associated with generalized eigenfunctions corresponding to the singularly continuous spectrum of B.