Browsing by Subject "Operator theory."

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Open Access
A measure disintegration approach to spectral multiplicity for normal operators
(2012) Ay, Serdar
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.
Open Access
Some criteria of selfadjointness for unbounded operators in Hilbert spaces
(2013) Özkaraca, Mustafa İsmail
This is a detailed presentation of some criteria of selfadjointness for unbounded operators in a Hilbert space, through operator Cauchy problems. We also include detailed preliminary results on unbounded linear operators in Hilbert spaces, the spectral theory of selfadjoint operators in Hilbert spaces, as well as the theory of extensions of Hermitian operators. The material of this thesis is classical, it was presented in the Operator Theory Seminar during the last two years, and contains material that can be found scattered through the textbooks cited in the bibliography list.