Spectra of linear fractional composition operators and properties of universal operators

• Main Author: Schroderus, Riikka
• Other Authors: Lindström, Mikael, University of Helsinki, Faculty of Science, Department of Mathematics and Statistics, Helsingin yliopisto, matemaattis-luonnontieteellinen tiedekunta, matematiikan ja tilastotieteen laitos, Helsingfors universitet, matematisk-naturvetenskapliga fakulteten, institutionen för matematik och statistik, Tylli, Hans-Olav, Nieminen, Pekka
• Format: Doctoral or Postdoctoral Thesis
• Language: English
• Published: Helsingin yliopisto 2017
• Subjects: matematiikka; Gutiérrez; Möbius; tylli
• Online Access: http://hdl.handle.net/10138/180931

The topics of this thesis in mathematics belong to the area of operator theory which, in general, studies linear transformations between complete normed vector spaces. Here, all operators considered are bounded and act on complex separable infinite-dimensional Hilbert space. A prototypical example of Hilbert spaces is formed by the square summable sequences of complex numbers. Other common Hilbert spaces consist of functions which are analytic on some open domain of the complex plane. The characteristic property of analytic functions is that they are locally given by a convergent power series and so the behaviour of such functions is rather rigid. Thesis consists of the introductory part and three research articles, the first and the third being co-authored with, respectively, E. A. Gallardo-Gutiérrez and H.-O. Tylli. Our focus in the first two articles is in the spectral properties of composition operators which are induced by linear fractional transformations (also known as Möbius maps). As the name suggests, a composition operator composes a function with a fixed mapping called the inducing map. In studying these operators we can take advantage of function theoretic tools, and it is not surprising that the properties of composition operator depend intricately on the inducing map. The spectrum of an operator acting on an infinite-dimensional space generalizes the concept of eigenvalues of a finite matrix. In general, determining the spectrum of a given operator is not an easy task. In the first article we compute the spectra of composition operators induced by certain linear fractional self-maps of the unit disc. Here the operators act on the whole range of weighted Dirichlet spaces which are Hilbert spaces of analytic functions on the unit disc. Earlier results in this context cover e.g. the classical Hardy space, the weighted Bergman spaces and the classical Dirichlet space. Our results complete the spectral picture of linear fractional composition operators on the weighted Dirichlet spaces. In particular, ...