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

Abstract

For a matrix-valued function $Φ\in L^2_{M_{n\times m}}$, it is well-known that the kernel of a block Hankel operator $H_Φ$ is an invariant subspace for the shift operator. Thus, if the kernel is nontrivial, then $\ker H_Φ= ΘH^2_{\mathbb C^r}$ for a natural number $r$ and an $m\times r$ matrix inner function $Θ$ by Beurling-Lax-Halmos Theorem. It will be shown that the size of the matrix inner function $Θ$ associated with the kernel of a block Hankel operator $H_Φ$ is closely related with a certain independency of the columns of $Φ$, 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.