Independence of Vector-Valued Functions Associated with Kernels of Block Hankel Operators

Abstract

For a block Hankel operator $$H_\Phi $$ with a matrix function symbol $$\Phi \in L^2_{M_{n\times m}}$$ , it is well-known that $$\ker H_\Phi = \Theta H^2_{\mathbb {C}^r}$$ for a natural number r and an $$m\times r$$ matrix inner function $$\Theta $$ if $$\ker H_\Phi $$ is nonempty. It will be shown that the size of the matrix inner function $$\Theta $$ associated with $$\ker H_\Phi $$ is closely related with a certain degree of independence of the column vectors of $$\Phi $$ , which will be defined in this paper. As an important application, the shape of shift invariant, or, backward shift invariant subspaces of $$H^2_{\mathbb {C}^n}$$ generated by finite elements will be studied.