INVARIANT SUBSPACES WITH THE CODIMENSION-ONE PROPERTY IN L(T)(MU)

In this paper, we refine a crucial lemma in [6] (Lemma 6) and give a s imple proof of it. Using this lemma, we prove that if p(infinity)(mu) = H-infinity(D) and D\spt mu has a component whose strong boundary has ' positive Lebesgue measure, then each invariant subspace of M of L(t) (mu) satisfies dim(M/zM) less than or equal to 1. The theorem is, in s ome senses, a generalization of Beurling's invariant subspace theorem for Hardy spaces. Let H-E(t) be the set of analytic functions on the o pen unit disk such that [GRAPHICS] where E is a closed subset of the u nit circle with positive Lebesgue measure. Let beta be a subset of L(a lpha)(t)(D) so that for each f is an element of beta there exists a bo unded analytic function phi satisfying phi f is an element of H-E(t). Using the theorem above, we show that if M is an invariant subspace of L(alpha)(t)(D) generated by beta and E satisfies the Carleson conditi on, then dim(M/zM) = 1. Finally, we prove that there exists an irreduc ible subnormal operator satisfying (1) sigma(S) = {z: \z\ less than or equal to 2}; (2) sigma(e)(S) = {z: \z\ = 1 or \z\ = 2}; and (3) Ind(S - lambda) = -1 for 1 < \lambda\ < 2 or \lambda\ < 1.