 # On the multicentric calculus for n-tuples of commuting operators

 dc.contributor Aalto-yliopisto fi dc.contributor Aalto University en dc.contributor.advisor Nevanlinna, Olavi dc.contributor.author Andrei, Diana dc.date.accessioned 2017-06-12T09:02:51Z dc.date.available 2017-06-12T09:02:51Z dc.date.issued 2017 dc.identifier.uri https://aaltodoc.aalto.fi/handle/123456789/26628 dc.description.abstract 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. en 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. dc.format.extent 122 dc.format.mimetype application/pdf en dc.language.iso en en dc.publisher Aalto University en dc.publisher Aalto-yliopisto fi dc.subject.other Mathematics en dc.title On the multicentric calculus for n-tuples of commuting operators en dc.type G3 Lisensiaatintutkimus fi dc.contributor.school Perustieteiden korkeakoulu fi dc.contributor.school School of Science en dc.contributor.department Matematiikan ja systeemianalyysin laitos fi dc.contributor.department Department of Mathematics and Systems Analysis en dc.subject.keyword multicentric calculus en dc.subject.keyword lemniscates en dc.subject.keyword von Neumann inequality en dc.subject.keyword Riesz projections en dc.subject.keyword commuting matrices en dc.identifier.urn URN:NBN:fi:aalto-201706125220 dc.type.dcmitype text en dc.programme.major Mathematics en dc.type.ontasot Licentiate thesis en dc.type.ontasot Lisensiaatintyö fi dc.contributor.supervisor Kinnunen, Juha dc.ethesisid Aalto 9197 dc.location P1
﻿