de Branges matrices and associated de Branges spaces of vector valued entire functions

TL;DR

This paper generalizes de Branges matrices to higher dimensions, explores associated vector-valued de Branges spaces, and introduces a parametrization and factorization methods for matrix-valued functions.

Contribution

It extends de Branges matrices to arbitrary even dimensions and develops a parametrization and factorization framework for matrix-valued entire functions.

Findings

Extended de Branges matrices to dimension 2n.

Developed a parametrization using the Smirnov maximum principle.

Discussed factorization of matrix-valued meromorphic functions.

Abstract

This paper extends the concept of de Branges matrices to any finite $m\times m$ order where $m=2n$. We shall discuss these matrices along with the theory of de Branges spaces of $\mathbb{C}^n$-valued entire functions and their associated functions. A parametrization of these matrices is obtained using the Smirnov maximum principle for matrix valued functions. Additionally, a factorization of matrix valued meromorphic functions is discussed.