Ay, Serdar2016-01-082016-01-082012http://hdl.handle.net/11693/15711Ankara : The Department of Mathematics and the Graduate School of Engineering and Science of Bilkent University, 2012.Thesis (Master's) -- Bilkent University, 2012.Includes bibliographical references leaves 44-45.In this thesis we studied the notion of direct integral Hilbert spaces, first introduced by J. von Neumann, and the closely related notion of decomposable operators, as defined in Kadison and Ringrose [1997] and Abrahamse and Kriete [1973]. Examples which show that some of the most familiar spaces in analysis are direct integral Hilbert spaces are presented in detail. Then we give a careful treatment of the notion of disintegration of a probability measure on a locally compact separable metric space, and using the machinery we obtain, a proof of the Spectral Multiplicity Theorem for Normal Operators employing the notion of disintegration of measures is given, based on Abrahamse and Kriete [1973], Arveson [1976], Arveson [2002]. In Chapter 5 the notion of essential preimage is presented in the sense of the article Abrahamse and Kriete [1973], and its relation with the spectral multiplicity function is discussed.vi, 45 leaves, 30 cmEnglishinfo:eu-repo/semantics/openAccessdirect integral Hilbert spacedisintegration of measuresnormal operatorsSpectral Multiplicity Theoremmultiplicity functionQA320 .A9 2012Spectral theory (Mathematics)Hilbert space.Functional analysis.Operator theory.Measure theory.Thesis