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 $\mathcal{H}(Δ) \equiv H_E^2 \ominus ΔH_{E}^2$, for some inner function $Δ$. Our first question calls for a description of the set $F$ in $H_E^2$ such that $\mathcal{H}(Δ)=E_F^*$, where $E_F^*$ 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 $L^2$-functions. Next, we ask: Is every shift-invariant subspace the kernel of a (possibly unbounded) Hankel operator? As we know, the kernel of a Hankel operator is shift-invariant, so the above question is equivalent to seeking a solution to the equation $\ker H_Φ^*=ΔH_{E^{\prime}}^2$, where $Δ$ is an inner function satisfying $Δ^* Δ=I_{E^{\prime}}$ almost everywhere on the unit circle $\mathbb{T}$ and $H_Φ$ denotes the Hankel operator with symbol $Φ$. Consideration of the above 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 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 (truncated backward shifts) between separable complex Hilbert spaces. In particular, we consider the multiplicity-free case.