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.
|Publication status||Published - 2017|
|MoE publication type||G3 Licentiate thesis|
- Multicentric calculus
- von Neumann inequality
- Riesz projections
- Commuting matrices