In operator theory, one of the central concepts is the spectrum of an operator and if one knows how to separate the spectrum into components, then the multicentric calculus is a useful tool, introduced by Olavi Nevanlinna in 2011.

This thesis presents extensions of the multicentric calculus from single operators to n-tuples of commuting operators, for both holomorphic and non-holomorphic functions. It also covers the same calculus when replacing the polynomials with rational functions. The multicentric representation of holomorphic functions gives a simple way to generalize the von Neumann result, i.e., the unit disc is a spectral set for contractions in Hilbert spaces. In other words, this calculus provides a way of representing the spectrum of a bounded operator T, by searching for a polynomial p that maps the spectrum to a small disc around origin. Since the von Neumann inequality works for contractions with spectrum in the unit disc, the multicentric representation applies a suitable polynomial p to the operator T, so that p(T) becomes a contraction with spectrum in the unit disc and thus the usual holomorphic functional calculus holds. When extending the calculus to n-tuples of commuting operators, a constant and some extra conditions are needed for the von Neumann inequality to hold true.

In order to extend the calculus to non-holomorphic functions, the Banach algebra is the tool to use in finding those functions for which it is possible to have a simple functional calculus by using suitable polynomial p. For a given bounded operator T on a Hilbert space, the polynomial p is such that p(T) is diagonalizable or similar to normal. The operators here are considered to be matrices. In particular, the calculus provides a natural approach to deal with non-trivial Jordan blocks. The extension of this calculus is done for a pair of commuting matrices, by constructing a suitable tensor product Banach algebra that can be identified with the space of continuous functions of two variables.

When replacing the polynomial by a rational function, one can apply the calculus to functions that are not polynomially convex, thus extending it to a larger class of functions.