The Hyperinvariant Subspace Lattice of a Contraction of Class C. 0

Abstract

It is shown that if T is a C.0 contraction with finite defect indices, then Hyperlat T is (lattice) generated by those subspaces which are either ker Ai(T) or ran i(T), where ip and J are scalar-valued inner functions. For a bounded linear operator T on a complex Hilbert space H, Hyperlat T denotes the lattice of all hyperinvariant subspaces for T, that is, the lattice of those subspaces which are invariant for all operators commuting with T. Recently, Fillmore, Herrero and Longstaff [1] showed that on a finite-dimensional space H, Hyperlat T is (lattice) generated by those subspaces which are either kerp(T) or ran q(T), where p and q are polynomials. In this note we generalize this to the following THEOREM. Let T be a contraction of class C.0 with finite defect indices acting on a separable Hilbert space. Then Hyperlat T is (lattice) generated by those subspaces which are either ker 4(T) or ran ~(T), where 4 and ( are scalarvalued inner functions. Recall that a contraction T (1 TIJ 6 1) is of class C.0 if T*nx -O0 for all x. The defect indices of T are, by definition, dT = rank(l-T*T)1/2 and dm, = rank(l TT*)1/2. If T is of class iC.0, then dT 6 dr,. For operators T, T' acting on H, H', respectively, T -< T' means that there exists a family of operators {Xa } from H to H' such that (i) for each a, Xax is one-to-one, (ii) Cl Ci VaXaH = H', and (iii) for each a, Xa T = T'X .If T < T' and T' < T, then T, T' are said to be completely injection-similar, and this is denoted by T c' T. For contractions of class C.0 and with dT = m < oo, d,. = n < oo, there has been developed a Jordan model which is, in a certain sense, analogous to the Jordan model for finite matrices. More specifically, if T is such a contraction then it is completely injection-similar to a uniquely determined Jordan operator of the form S((p1) D e * * D S(pk) ED Sn, Received by the editors July 19, 1977 and, in revised form, January 25, 1978. AMS (MOS) subject classifications (1970). Primary 47B99; Secondary 47A15.