Multicentric calculus and the Riesz projection

TL;DR

This paper advances multicentric holomorphic calculus by addressing polynomial lemniscates for spectral set separation and extending the calculus to handle root multiplicities, improving spectral projection computations.

Contribution

It introduces methods for separating spectral components with polynomial lemniscates and incorporates root multiplicities into multicentric calculus.

Findings

Enhanced techniques for spectral set separation using polynomial lemniscates.

Modified calculus framework to include multiple roots.

Applications to computing Riesz spectral projections.

Abstract

In multicentric holomorphic calculus one represents the function $\varphi$ using a new polynomial variable $w=p(z)$ in such a way that when it is evaluated at the operator $A,$ then $p(A)$ is small in norm. Usually it is assumed that $p$ has distinct roots. In this paper we discuss two related problems, the separation of a compact set (such as the spectrum) into different components by a polynomial lemniscate, respectively the application of the Calculus to the computation and the estimation of the Riesz spectral projection. It may then become desirable the use of $p(z)^n$ as a new variable. We also develop the necessary modifications to incorporate the multiplicities in the roots.