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Abstract:
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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. |