On Toeplitz operators which are contractions

Abstract

We prove that a Toeplitz contraction T, is completely nonunitary if q is not a constant. As an application, it is noted that for such Tp, a functional calculus can be defined for all functions u in HOO of the unit disk. For I _p_ ?O, we denote by LP the usual class of Lebesgue measurable functions on the unit circle y of the complex plane. We write HP for the closed subspace of LP of functions whose Fourier coefficients vanish on the negative integers. We denote by P the orthogonal projection of L2 onto H2 and by B(H2) the space of bounded operators on H2. For b E LO, we consider the Toeplitz operator T,k eB(H2) defined by TJkf=P(Qf) for f E H2. Following Sz.-Nagy and Foias [2], we say a contraction TP, 11 T,ll 0 and feS. We may apply the F. and M. Riesz theorem [1, p. 82] to the equality II Xf 112= !If 11 2 for a nonzero f e S to conclude that I kl = 1 almost everywhere on y. Thus, we write S={f e H2:+bf, bnfe H2 for n_O}. Let Mz be the operator of multiplication by the coordinate function z. Then bnf e H2 implies z bnf=n zf E H2 and similarly for Xnzf i.e. M,Sc S. By Beurling's theorem [1, p. 79], there is a function Vp E HO, IVy=1 almost everywhere, such that S=VpH2. Since 1 eH2, cp is in S. Note that f?p = pf for some f E H2 (since b?p e S=?pH2). Hence for Received by the editors November 15, 1971. AMS 1970 subject classifications. Primary 47B35, 47A20, 47A60; Secondary 46J15.