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

In Sung Hwang, Woo Young Lee

Published 2018-05-17Version 1

In this paper we 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 the strong $L^2$-function is of bounded type. On the other hand, the kernel of a Hankel operator is shift-invariant. Thus in view of its converse, we may ask whether every shift-invariant subspace is represented by the kernel of a Hankel operator. This question invites us to consider a solution of the equation involved with the unbounded Hankel operators corresponding to the given inner function. In this context, we introduce a notion of the "Beurling degree" for inner functions by employing the 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. In addition, we investigate the case of Beurling degree 1.