Controlled factorisation for some commuting pairs of contractions with thin spectrum
by
Andrei Halanay
University Politehnica of Bucharest

For F, F_* separable Hilbert spaces and \Theta in H^\infty (D², L (F, F_*)), ||\Theta||_\infty <= <= 1, \Theta left-inner, define H = H_\Theta = H² (D², F_*) \ominus\ThetaH² (D², F) and then A = P_H M_z|_H, T = P_H M_\xi|_H with M_z, M_\xi operators of multiplication with independent variables.

Suppose \sigma_re (\Theta) = {(\lambda, \mu) in D² | exists{e_n}_n an orthonormal sequence in F_* such that lim_{n --> \infty} ||\Theta(\lambda, \mu)^* e_n || = 0} dominates \gamma×\gamma with \gamma subset T, 1 in \gamma.

An w^*-closed subspace of H^\infty (D²), E, is associated to \gamma×\gamma and for each \lambda = (\lambda₁, \lambda₂) in D² one can find x, y in H such that [x·y] = [P_\lambda] in Q_E and x·y satisfies also some growth conditions. An approach to existence of joint nontrivial invariant subspaces for A and T follows from this, similar to results from single operator case.

Date received: June 13, 2002