ONE DIMENSIONAL PERTURBATIONS OF RESTRICTED SHIFTS* By DOUGLAS N. CLARK in Los Angeles, California, U.S.A. 1. Introduction. The purpose of this paper is to initiate the study of unitary operators which are one dimensional perturbations of unilateral shift operators, restricted to certain of their *-invariant subspaces. More precisely, let H 2 denote the Hardy class of the unit disk and M a subspace of H 2 invariant under multiplication by z. We study the unitary operators (denoted Uw, Iwl-- 1) which are one dimensional perturbations of the operator S:M-L = HZ@M ~ M • defined by Sf = PMJ_zf; PM" the projection on M -L. The origin of our study is the question of determining conditions on a se- quence ct, c2,.., in the unit disk which insure that the functions t kz(z) = (1 - fo(2)fo(Z))/(1 - ,~z), 2 = cl, c2, "" span M -L, where fo(z) is the inner function which generates M: M = fo H2. A special case of this problem is that in which fo(z) = exp[--(1 + z)/(1 -- z)]. * This work was partially supported by N.S.F. Grant GP-11475. 1 f(z) denotes the conjugate off(z):f(z) = f(z--). 169