Kernels of block Hankel operators and independency of vector-valued functions modulo Nevanlinna class

TL;DR

This paper explores the relationship between the kernel of block Hankel operators and the independence of vector-valued functions modulo Nevanlinna class, with applications to shift-invariant subspaces.

Contribution

It establishes a connection between the size of the inner function in the kernel and the independence of the columns of the symbol function, advancing understanding of invariant subspaces.

Findings

Kernel size relates to column independence of the symbol function.

Characterization of shift-invariant subspaces generated by finite elements.

Application to the structure of backward shift invariant subspaces.

Abstract

For a matrix-valued function $\Phi\in L^2_{M_{n\times m}}$, it is well-known that the kernel of a block Hankel operator $H_\Phi$ is an invariant subspace for the shift operator. Thus, if the kernel is nontrivial, then $\ker H_\Phi= \Theta H^2_{\mathbb C^r}$ for a natural number $r$ and an $m\times r$ matrix inner function $\Theta$ by Beurling-Lax-Halmos Theorem. It will be shown that the size of the matrix inner function $\Theta$ associated with the kernel of a block Hankel operator $H_\Phi$ is closely related with a certain independency of the columns of $\Phi$, which is defined in this paper. As an important application of this result, the shape of shift invariant, or, backward shift invariant subspaces of $H^2_{\mathbb C^n}$ generated by finite elements will be studied.