### A representation of the minimal $P$-norm solution.

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In this paper we construct a few iterative processes for computing $\left\{2\right\}$-inverses of a linear bounded operator. These algorithms are extensions of the corresponding algorithms introduced in [11] and a method from [8]. A few error estimates are derived.

We investigate the generalized Drazin inverse and the generalized resolvent in Banach algebras. The Laurent expansion of the generalized resolvent in Banach algebras is introduced. The Drazin index of a Banach algebra element is characterized in terms of the existence of a particularly chosen limit process. As an application, the computing of the Moore-Penrose inverse in ${C}^{*}$-algebras is considered. We investigate the generalized Drazin inverse as an outer inverse with prescribed range and kernel....

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