Some invariant subspaces for A-contractions and applications

Abstract

Some invariant subspaces for the operators A and T acting on a Hilbert space H and satisfying T *AT ≤ A and A ≥ 0, are presented. Especially, the largest invariant subspace for A and T on which the equality T *AT = A occurs, is studied in connections to others invariant or reducing subspaces for A, or T . Such subspaces are related to the asymptotic form of the subspace quoted above, this form being obtained using the operator limit of the sequence {T *ⁿAT ⁿ; n ≥ 1}. More complete results are given in the case when AT = A¹̸²TA¹̸ ². Also, several applications for quasinormal operators are derived, involving their unitary, isometric and quasi-isometric parts, as well as their asymptotic behaviour.