Multicentric calculus and the Riesz projection

Abstract

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