A canonical decomposition of strong $L^2$-functions

TL;DR

This paper introduces a canonical decomposition for strong $L^2$-functions using the Beurling-Lax-Halmos Theorem, linking spectral multiplicity with a new concept called Beurling degree.

Contribution

It presents a novel decomposition method for operator-valued $L^2$-functions and defines the Beurling degree, connecting spectral properties with inner functions.

Findings

Established a canonical decomposition for strong $L^2$-functions.

Introduced the concept of Beurling degree of inner functions.

Connected spectral multiplicity with Beurling degree.

Abstract

The aim of this paper is to establish a canonical decomposition of operator-valued strong $L^2$-functions by the aid of the Beurling-Lax-Halmos Theorem which characterizes the shift-invariant subspaces of vector-valued Hardy space. This decomposition invites us to coin a new notion of the "Beurling degree" of the inner function. Eventually, we establish a deep connection between the spectral multiplicity of the model operator and the Beurling degree of the corresponding characteristic function.