Let A be a bounded self-adjoint operator on a separable Hilbert
space H and H0ββH a closed invariant
subspace of A. Assuming that H0β is of codimension 1,
we study the variation of the invariant subspace H0β under bounded
self-adjoint perturbations V of A that are off-diagonal
with respect to the decomposition H=H0ββH1β. 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