The Beurling-Lax-Halmos theorem for infinite multiplicity

Abstract

In this paper we consider several questions emerging from the Beurling-Lax-Halmos Theorem, which characterizes the shift-invariant subspaces of vector-valued Hardy spaces. The Beurling-Lax-Halmos Theorem states that a backward shift-invariant subspace is a model space H(Δ)≡HE2⊖ΔHE2, for some inner function Δ. Our first question calls for a description of the set F in HE2 such that H(Δ)=EF⁎, where EF⁎ denotes the smallest backward shift-invariant subspace containing the set F. In our pursuit of a general solution to this question, we are naturally led to take into account a canonical decomposition of operator-valued strong L2-functions. Next, we ask: Is every shift-invariant subspace the kernel of a (possibly unbounded) Hankel operator? Consideration of the question on the structure of shift-invariant subspaces leads us to study and coin a new notion of “Beurling degree” for an inner function. We then establish a deep connection between the spectral multiplicity of the model operator (the truncated backward shift) and the Beurling degree of the corresponding characteristic function. At the same time, we consider the notion of meromorphic pseudo-continuations of bounded type for operator-valued functions, and then use this notion to study the spectral multiplicity of model operators between separable complex Hilbert spaces. In particular, we consider the case of multiplicity-free: more precisely, for which characteristic function Δ of the model operator T does it follow that T is multiplicity-free, i.e., T has multiplicity 1? We show that if Δ has a meromorphic pseudo-continuation of bounded type in the complement of the closed unit disk and the adjoint of the flip of Δ is an outer function, then T is multiplicity-free.