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 ? R# ? R+. Then we firstly show that an element a ? REP if and only if and Equation : axa+ + a+ax = 2x has at least one solution in ?a = {a, a#, a+, a+, (a#)+, (a+)+}. Next, a ? RSEP if and only if Equation: axa+ + a+ax = 2x has at least one solution in ?a. Finally, a ? RPI if and only if Equation: aya+x = xy has at least one solution in ?2a , where ?a = {a, a#, a+, (a#)+, (a+)+}.
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