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

Abstract

For a matrix-valued function $\Phi \in L_{M_{n\times m}}^2$, 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 $\ker H_{\Phi }=\Theta H_{{\mathbb{C}}^r}^2$ 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 $H_{{\mathbb{C}}^n}^2$ generated by finite elements will be studied.