Multicentric calculus and the Riesz projection

In multicentric holomorphic calculus one represents the function φ 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.