On m-generalized invertible operators on Banach spaces

Abstract

A bounded linear operator $S$ on a Banach space $X$ is called an $m$-left generalized inverse of an operator $T$ for a positive integer $m$ if

$$T{\sum }_{j=0}^m(-1{)}^j(\begin{array}{c}\hfill m\hfill \\ \hfill j\hfill \end{array})S^{m-j}{T}^{m-j}=0,$$ and it is called an $m$-right generalized inverse of $T$ if

$$S{\sum }_{j=0}^m(-1{)}^j(\begin{array}{c}\hfill m\hfill \\ \hfill j\hfill \end{array})T^{m-j}{S}^{m-j}=0.$$ If $T$ is both an $m$-left and an $m$-right generalized inverse of $T$, then it is said to be an $m$-generalized inverse of $T$.

This paper has two purposes. The first is to extend the notion of generalized inverse to $m$-generalized inverse of an operator on Banach spaces and to give some structure results. The second is to generalize some properties of $m$-partial isometries on Hilbert spaces to the class of $m$-left generalized invertible operators on Banach spaces. In particular, we study some cases in which a power of an $m$-left generalized invertible operator is again $m$-left generalized invertible.