On the Reflextvtty of C0(n) Contractions

Let T be a CQ(N) contraction on a separable Hubert space and let / — S(qPi) ffi S( 1 if there exists an inner function (T) = 0 and the defect indices of T, dT = rank(/ T* T)x/2 and dj, = rank(/ 7T*)1/2, are both equal to some M < N. A C0(N) contraction is unitarily equivalent to the operator T defined on H = HJjQ &TH2; by Tf = P(e"f) for f E H, where H¡, denotes the standard Hardy space of C^-valued functions defined on the unit circle, 6r is the characteristic function of T, and P denotes the (orthogonal) projection from Hf¿ onto H (cf. [5, Chapter VI]). Two operators Tx, T2 are quasi-similar if there exist one-to-one operators X and Y with dense ranges (called quasi-affinities) such that AT, = T2X and YT2 = Tx Y. A C0(N) contraction is quasi-similar to a uniquely determined Jordan operator (called its Jordan model) J = S(j) denotes the operator defined on H2 0 tpjH2 by S(yj)f = Pj(euf) for/ G >Y2 © jH2, j = 1, 2, . . . , k (cf. [4]). For £ and ij in HM, £ /\ tj = 1 denotes that £ and 17 have no nontrivial common inner divisor. Received by the editors February 6, 1979 and, in revised form, July 11, 1979. AMS (MOS) subject classifications (1970). Primary 47A45; Secondary 47A15.