On restrictions of operators

Abstract

Introduction. Let r be a bounded linear operator on a Banach space B and let r-1 exist. Consider a closed subspace C of B which is invariant under r, i.e. rCCC. The restriction of r to C is an operator T on the Banach space C. Suppose now that T-'CCC, so that T has no inverse. The class of operators T arising in this way has rather special properties, which are in general quite different from those of r on, the whole space B. The purpose of this paper is to study this class of restriction operators T. Throughout we shall make one simplifying assumption on C, namely, that C contains a single x, i.e. a vector x such that the vectors Tx, n =0, 1, 2, . .. , are fundamental in C. It is clear that if r-'CgC and x is a generator of C, then r-lx C. In order that for a given operator r there may exist an invariant subspace C which is not invariant under r-1, it is necessary that the resolvent set of r have more than one component. For let x be in C and take any functional i& with +t(C) =0. Then 4,b((XI-r)-1x) = 0 ('rnx, 15&)/Xn+l=O for large IXl and hence for the entire resolvent set if this set is connected. In particular, 4I(r-'x)=0; thus (C)=0 implies(,(r-lx) =O and so r7-xEC. Hence r-CC C. In ?1 we shall show how an operator T on C, obtained by restricting T to C, can be represented as an operator of multiplication by z on a space of analytic functions. We shall also investigate the ring of operators on C which commute with T. In ?2 we shall show how these general considerations specialise in the case of certain normal operators on Hilbert space. In particular, we shall discuss some results of Kolmogoroff on stationary sequences which correspond in our problem to the case of a unitary operator r on Hilbert space.