Some characterizations of strongly partial isometry elements in rings with involutions

Abstract

In this paper, we study an element which is both group invertible and Moore penrose invertible to be EP, partial isometry and strongly EP by discussing the existence of solutions in a definite set of some given constructive equations. Mainly, let $a \in R^\# \cap R^+$. Then we first show that an element $a \in R^{EP}$ if and only if and Equation $: $ $axa^++a^+ax=2x$ has at least one solution in $\chi_a =\{a, a^\#, a^+, a^\ast, (a^\#)^\ast, (a^+)^\ast\}$. Next, $a \in R^{SEP}$ if and only if Equation: $ axa^\ast+a^+ax=2x$ has at least one solution in $\chi_a$. Finally, $a \in R^{PI}$ if and only if Equation: $aya^{*}x=xy$ has at least one solution in $\rho^{2}_{a}$, where $\rho_{a}=\{a, a^\#, a^+, (a^\#)^*, (a^+)^*\}$.