Browsing Mathematics by Author "Ivković, Stefan"
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Ivković, Stefan (Beograd , 2021)[more][less]
Abstract: In the first part of the thesis, we establish the semi-Fredholm theory on Hilbert C∗- modules as a continuation of the Fredholm theory on Hilbert C∗-modules which was introduced by Mishchenko and Fomenko. Starting from their definition of C∗-Fredholm operator, we give definition of semi-C∗-Fredholm operator and prove that these operators correspond to one-sided invertible elements in the Calkin algebra. Also, we give definition of semi-C∗-Weyl operators and semi-C∗-B-Fredholm operators and obtain in this connection several results generalizing the counterparts from the classical semi-Fredholm theory on Hilbert spaces. Finally, we consider closed range operators on Hilbert C∗-modules and give necessary and sufficient conditions for a composition of two closed range C∗-operators to have closed image. The second part of the thesis is devoted to the generalized spectral theory of operators on Hilbert C∗-modules. We introduce generalized spectra in C∗-algebras of C∗-operators and give description of such spectra of shift operators, unitary, self-adjoint and normal operators on the standard Hilbert C∗- module. Then we proceed further by studying generalized Fredholm spectra (in C∗-algebras) of operators on Hilbert C∗-modules induced by various subclasses of semi-C∗-Fredholm operators. In this setting we obtain generalizations of some of the results from the classical spectral semi-Fredholm theory such as the results by Zemanek regarding the relationship between the spectra of an operator and the spectra of its compressions. Also, we study 2×2 upper triangular operator matrices acting on the direct sum of two standard Hilbert C∗-modules and describe the relationship between semi-C∗-Fredholmness of these matrices and of their diagonal entries. URI: http://hdl.handle.net/123456789/5305 Files in this item: 1
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