The Schur transformation for Nevanlinna functions: operator representations, resolvent matrices, and orthogonal polynomials

TL;DR

This paper introduces a fractional linear transformation for Nevanlinna functions, analyzing its impact on their operator representations, resolvent matrices, and orthogonal polynomial structures.

Contribution

It develops a Schur-like transformation for Nevanlinna functions and studies its effects on their operator and resolvent representations.

Findings

Transformation preserves certain operator structures

Explicit relations between original and transformed functions

Applications to orthogonal polynomials and resolvent matrices

Abstract

A Nevanlinna function is a function which is analytic in the open upper half plane and has a non-negative imaginary part there. In this paper we study a fractional linear transformation for a Nevanlinna function $n$ with a suitable asymptotic expansion at $\infty$, that is an analogue of the Schur transformation for contractive analytic functions in the unit disc. Applying the transformation $p$ times we find a Nevanlinna function $n_p$ which is a fractional linear transformation of the given function $n$. The main results concern the effect of this transformation to the realizations of $n$ and $n_p$, by which we mean their representations through resolvents of self-adjoint operators in Hilbert space. Our tools are block operator matrix representations, $u$--resolvent matrices, and reproducing kernel Hilbert spaces.