Rational functions as new variables

TL;DR

This paper explores using rational functions as new variables in multicentric calculus, enabling representation of functions near non-polynomially convex sets and deriving new power series solutions for operator equations.

Contribution

It introduces modifications to multicentric calculus by replacing polynomial variables with rational functions, broadening the scope of functions and domains that can be analyzed.

Findings

Power series representation for Sylvester equations with bounded operators.

K-spectral results for bounded operators in Hilbert spaces.

Extension of multicentric calculus to rational functions.

Abstract

In multicentric calculus one takes a polynomial $p$ with distinct roots as a new variable and represents complex valued functions by $\mathbb C^d$-valued functions, where $d$ is the degree of $p$. An application is e.g. the possibility to represent a piecewise constant holomorphic function as a convergent power series, simultaneously in all components of $|p(z)| \le \rho$. In this paper we study the necessary modifications needed, if we take a rational function $r=p/q$ as the new variable instead. This allows to consider functions defined in neighborhoods of any compact set as opposed to the polynomial case where the domains $|p(z)| \le \rho$ are always polynomially convex. Two applications are formulated. One giving a convergent power series expression for Sylvester equations $AX-XB =C$ in the general case of $A,B$ being bounded operators in Banach spaces with distinct spectra. The other application formulates a K-spectral result for bounded operators in Hilbert spaces.