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

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 reduces to the Douglas-Shapiro-Shields factorization if the flip of a strong $L^2$-function is of bounded type. To consider a converse of the Beurling-Lax-Halmos Theorem,we introduce a notion of the "Beurling degree" for inner functions by employing a canonical decomposition of strong $L^2$-functions induced by the given inner functions. Eventually, we establish a deep connection between the Beurling degree of the given inner function and the spectral multiplicity of the truncated backward shift on the corresponding model space.