Wandering Property in the Hardy Space

摘要

Let X be a Hilbert space and let V: X → X be a bounded linear operator. If V is an isometry, then the well-known Wold decomposition theorem states that X = X_0 ⊕From n=0 to ∞ of V~n X_1, (1) where X_1 = X direct- VX is a wandering subspace and X_0 = ∩ from n=0 to ∞ of V~nX. If X = H~2 and V is the operator of multiplication by an inner function g, then the intersection ∩ from n=0 to ∞ of V~nH~2 = {0} and the decomposition implies that an orthonormal basis of H~2 direct- gH~2, {s_1,..., s_n,...}, is a g-basis of H~2; that is, any function f ∈ H~2 can be written as f(2) = ∑ From n=0 to ∞ of s_n(z)f_n(g(z)). (2) Any closed subspace M is contained in H~2 that is invariant under multiplication by g could be considered as X, and therefore a relation similiar to (2) holds. We write this relation in the following form.