MULTIPLICATION INVARIANT SUBSPACES OF HARDY-SPACES

This paper studies closed subspaces L of the Hardy spaces H-p which ar e g-invariant (i.e.,g . L subset of or equal to L) where g is inner, g not equal 1. If p = 2, the Wold decomposition theorem implies that th ere is a countable ''g-basis'' f(1),f(2),... of L in the sense that L is a direct sum of spaces f(j) . H-2[g] where H-2[g] = {f o g \ f is a n element of H-2}. The basis elements f(j) satisfy the additional prop erty that integral(T)\f(j)\(2)g(k) = 0, k = 1, 2,.... We call such fun ctions g-2-inner. It also follows that any f is an element of H-2 can be factored f = h(f,2).(F-2 o g) where h(f,2) is g-2-inner and F is ou ter, generalizing the classical Riesz factorization. Using L(p) estima tes for the canonical decomposition of H-2, we find a factorization f = h(fp).(F-p o g) for f is an element of H-P. If p greater than or equ al to 1 and g is a finite Blaschke product we obtain, for any g-invari ant L subset of or equal to H-P, a finite g-basis of g-p-inner functio ns.