On operators on Hilbert $C^*$-modules

We will present some recent results in the operator theory on Hilbert $C^*$-modules. Motivated by Mishchenko`s definition of Fredholm operator on Hilbert C*-module, we introduce definition of semi-Fredholm operator on Hilbert and also define several new classes of operators on Hilbert $C^*$-modules as a generalization of semi-Weyl operators on Hilbert spaces. Working with these new classes of operators, we obtain generalizations of many results from the classical semi-Fredholm and semi-Weyl theory on Hilbert spaces. We consider both adjointable and general bounded operators over $C^*$-algebras. The special case of operators over $W^*$-algebras is also considered, which allows us to generalize several results from the classical semi-Fredholm theory than in the case of arbitrary C*-algebras. Also, we introduce concrete examples of such operators and illustrate how they may differ from the classical semi-Fredholm operators on Hilbert spaces.
Next, we study closed range operators on Hilbert $C^*$-modules and give necessary and sufficient conditions for that a composition of two closed range operators to have closed image.
We also introduce the generalized spectra in $C^*$-algebras of operators on Hilbert $C^*$-modules and give a description of such spectra for shift operators, unitary, self-adjoint and normal operators on Hilbert $C^*$-modules. Further, we obtain a generalization in this setting of the results by Zemanek and some other results from the classical spectral semi-Fredholm theory.
Finally, we study linear dynamics of elementary mappings on the space of compact operators on Hilbert $C^*$-modules and give necessary and sufficient conditions for such maps to be topologically transitive or chaotic. We also give examples of the mappings satisfying these conditions.
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