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

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.

This paper was incorporated in a part of the new paper [The Beurling-Lax-Halmos Theorem for Infinite Multiplicity; arXiv:1910.09957] co-authored with Raul E. Curto