弱收縮變換之雙不變子空間

Abstract

For a bounded linear operator T acting on a complex, separable Hilbert space, let Lat T,Lat''T and Hyperlat T denote the lattices of invariant subspace, bi-invariant subspaces and hyperinvariant subspaces of T, respectively. In this paper we characterize the elements of Lat''T, in terms of the characteristic function of T, when T is a completely non-unitary weak contraction with finite defect indices. We show that if the defect indices of T are n <= and ΘT denotes the characteristic function of T , then a subspace in Lat T belongs to Lat''T if and only if the intermediate space of its corresponding regular factorization ΘT=Θ2 Θ1 is of dimension n.As corollaries, necessary and sufficient conditions that two of these lattices of subspaces be equal to each other are obtained.In particular, if T1, T2 are completely non-unitary C11 contractions with finite defect indices which are quasi-similar to each other, then Lat''T1 is isomorphic to Lat "T2.Whether this is true for weak contractions is still unknown.