Extensions of the multicentric functional calculus

Abstract

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.