The Beurling-Lax-Halmos Theorem for Infinite Multiplicity

TL;DR

This paper extends the Beurling-Lax-Halmos Theorem to infinite multiplicity, exploring shift-invariant subspaces, Hankel operators, and introducing the concept of Beurling degree to connect spectral multiplicity with model operator characteristics.

Contribution

It introduces the notion of Beurling degree for inner functions and links spectral multiplicity of model operators to this new concept, advancing the understanding of shift-invariant subspaces.

Findings

Characterization of shift-invariant subspaces via a canonical decomposition.

Establishment of a connection between Beurling degree and spectral multiplicity.

Analysis of meromorphic pseudo-continuations for operator-valued functions.

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 $\mathcal{H}(\Delta) \equiv H_E^2 \ominus \Delta H_{E}^2$, for some inner function $\Delta$. Our first question calls for a description of the set $F$ in $H_E^2$ such that $\mathcal{H}(\Delta)=E_F^*$, 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 $\ker H_{\Phi}^*=\Delta H_{E^{\prime}}^2$, where $\Delta$ is an inner function satisfying $\Delta^* \Delta=I_{E^{\prime}}$ almost everywhere on the unit circle $\mathbb{T}$ and $H_{\Phi}$ denotes the Hankel operator with symbol $\Phi$. 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.