On the multicentric calculus for n-tuples of commuting operators 

One of the central concepts of operator theory is the spectrum of an operator and if one knows that the spectrum is separated then the multicentric calculus is a useful tool introduced by Olavi Nevanlinna in 2011. This thesis is an attempt of extending the multicentric calculus from single operators to n-tuples of commuting operators, both for holomorphic functions and for nonholomorphic 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. The multicentric calculus without assuming the functions to be analytic provides a way to construct a Banach algebra, depending on the polynomial p, for which a simple functional calculus holds. 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 nontrivial Jordan blocks. In the attempt to extend this calculus to n-tuples of commuting matrices, formulas only for cases when n=2 and n=3 are provided because of the complexity and length of a more general formula.