98 research outputs found
The singularly continuous spectrum and non-closed invariant subspaces
Let be a bounded self-adjoint operator on a separable Hilbert
space and a closed invariant
subspace of . Assuming that is of codimension 1,
we study the variation of the invariant subspace under bounded
self-adjoint perturbations of that are off-diagonal
with respect to the decomposition . In particular, we prove the existence of a
one-parameter family of dense non-closed invariant subspaces of the operator
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
- β¦