Generalized Type Kato Operator And Weyl Type Theorem

In this paper, we give the definition of generalized Kato type operators and de-fine the new spectrum by the property of generalized Kato type operators. Then by means of the spectra induced in view of the operators of consistency in invertibility and in the Fredholm and index, we establish the sufficient and necessary conditions for a bounded linear operator for which Weyl type theorem and its perturbation hold. Finally, the theory is exemplified in hypercyclic operators and H(P) operators.This paper contains three chapters:In chapter1, we give the definition of generalized Kato type operators and define the new spectrum by the property of generalized Kato type operators. Then by means of the spectra induced in view of the operators of consistency in invertibility and in the Fredholm and index, we establish the sufficient and necessary conditions for a bounded linear operator for which property (ω) and its perturbation hold. Finally, as an application of the main results, we study the relationship between property (ω) and hypercyclic operators.In chapter2, by means of the spectra defined in view of generalized Kato prop-erty and the property of consistence in Fredholm and index, we give the sufficient and necessary conditions for a bounded linear operator for which generalized prop-erty (ω) holds. In addition, we obtain the sufficient and necessary conditions for a bounded linear operator which has generalized property (ω) under perturbations by finite rank and power finite rank operators.In chapter3, in view of the spectra defined in chapter2, we give the suffi-cient and necessary conditions for a bounded linear operator for which generalized property (ω) holds. Meanwhile, the sufficient and necessary conditions for which the generalized property (ω’) of operational calculus are given. Furthermore, as a consequence of the main results, the stability of property (ω’) is discussed. Finally, the theory is exemplified in H(P) operators.