Jakob Reiffenstein: Interlacing zeroes and poles of matrix-valued meromorphic Herglotz-Nevanlinna functions

Abstract:

This talk is about meromorphic Herglotz-Nevanlinna functions, i.e., functions that are meromorphic on all of \(\mathbb{C}\), real-valued on \(\mathbb{R}\) and whose values in the upper half-plane have non-negative imaginary part. They have the characteristic property that their zeroes and poles are all real, simple, and interlacing: between any two zeroes there is a pole and between any two poles there is a zero. Conversely, any meromorphic function that is real-valued on \(\mathbb{R}\) and has real, simple, and interlacing zeroes and poles is a Herglotz-Nevanlinna function if it does not grow too fast.

Now consider an \((n \times n)\)-matrix-valued meromorphic Herglotz-Nevanlinna function, with the goal of establishing a similar interlacing criterion. One needs to find the right interpretation of “zeroes and poles”, and give a suitable generalisation of the term “interlacing”. I will present a notion of generalized interlacing, as well as the following result: A matrix-valued meromorphic function, whose values on \(\mathbb{R}\) are Hermitian matrices, is Herglotz-Nevanlinna if and only if it satisfies an interlacing condition and does not grow too fast.