Certain invariant subspace structure of $L^2(\Bbb T^2)$ II

Abstract

Let M be an invariant subspace of L2(T2). Considering the largest z- invariant (resp. w-invariant) subspace Fz (resp. Fw) in the wandering subspace M zwM of M with respect to the shift operator zw. If Fw 6 f0g and Fz 6 f0g, then we consider the certain form of invariant subspaces M of L2(T2). Furthermore, we study certain classes of invariant subspaces of L 2 (T 2 ). 1. Introduction and preliminaries Let T 2 be the torus that is the cartesian product of 2 unit circles in C. Let L 2 (T 2 ) and H 2 (T 2 ) be the usual Lebesgue and Hardy space on the torus T 2 , respectively. A closed subspace M of L 2 (T 2 ) is said to be invariant if zM ‰ M and wM ‰ M. As is well known, the structure of invariant subspaces is much more complicated. In general, the invariant subspaces of L 2 (T 2 ) are not necessarily of the form φH 2 (T 2 ) with some unimodular function φ. The structure of Beurling-type invariant subspaces has been studied, and some necessary and sufficient conditions for invariant subspaces to be Beurling-type have been given (cf. (1, 2, 5), etc). Further, many authors had attempted to study the form of invariant subspaces of L 2 (T 2 ) (cf. (4, 6, 7), etc). In (4), we studied the structure of an invariant subspace M as a zw- invariant subspace. We gave an alternative approach of Beuring-type in- variant subspaces and a certain class of invariant subspace which contains the class of invariant subspaces of the form φH 2 0 (T 2 ), where H 2 0 (T 2 ) = ff 2 H 2 (T 2 ) : f (0, 0) = 0g and φ is a unimodular function in L 1 (T 2 ). For (m, n) 2 Z 2 and f 2 L 2 (T 2 ), the Fourier coefficient of f is defined