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The paper introduces the left and right versions of the large class of Drazin inverses in terms of the left and right annihilators in a ring, which are called left-Drazin and right-Drazin inverses. We characterize some basic properties of these one-sided Drazin inverses, and discuss Jacobson's lemma for them. In addition, the relation between the Drazin inverses and these two one-sided inverses is given by means of the spectrum and the operator decomposition. As an application, the left-Drazin and right-Drazin invertibilities in the Calkin algebra are investigated. [ABSTRACT FROM AUTHOR]

Based on the concept and properties of $C^{\ast}$-algebras, the paper introduces a concept of $C^{\ast}$-algebra-valued b-metric spaces which generalizes the concept of $C^{\ast}$-algebra-valued metric spaces and gives some basic fixed point theorems for self-map with contractive condition on such spaces. As applications, existence and uniqueness results for a type of operator equation and an integral equation are given. [ABSTRACT FROM AUTHOR]

We give the necessary and sufficient condition for a bounded linear operator with property ( ω) by means of the induced spectrum of topological uniform descent, and investigate the permanence of property ( ω) under some commuting perturbations by power finite rank operators. In addition, the theory is exemplified in the case of algebraically paranormal operators. [ABSTRACT FROM AUTHOR]