  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.
