Several Characterizations Of Partial Order Equidistant Elements

Generalized inverses are widely used in many fields,and a number of scholars extensively studied generalized inverses of complex matrices,Banach algebras,bounded linear operators on Banach spaces,C*-algebras,rings and semigroups and obtained remarkable results.Because it is difficult to study the generalized inverses of a ring,many problems remain to be discussed.To highlight the new perspective of studying partial isometries,the paper mainly focuses on characterizations of partial isometries in a ring with involution,and the equivalent relation between partial isometries and solutions of specific equation by the theory of the ring.It mainly consists of three parts.The first part briefly summarizes the origin and development of generalized inverses,including the application prospect of generalized inverses in production practice.At the same time.some basic concepts of generalized inverses in a ring,basic knowledge and relevant lemmas to be used in this paper are also presented.The second part mainly focuses on characterizations of partial isometries under some conditions.First of all,we discuss the equivalent conditions for an element in a ring to be a partial isometry,which is MP-invertible.For example,let a?R+,then a is a partial isometry if and only if a=aa*a or a*=a*aa*.In addition,we give the equivalent conditions for an element in a ring to be a partial isometry involving powers of nilpotent elements,idempotents in ring R and Jacobson radicals of the ring R.Such as,let a?R+,then a is a partial isometry if and only if aa*is a idempotent and aa*-aa+ is a nilpotent element.Then,we present some characterizations for an element in a ring to be a partial isometry,which is MP-invertible and group invertible.For instance,let a ?R#? R+,then a is a partial isometry if and only if a*a+= a+a+or a+a*=a+a+,which enriched Mosic and Djordjevic's conclusions.In third part,we construct some corresponding equations by analyzing characterizations of partial isometries and consider the equivalence between an element in a ring to be a partial isometry and the solutions of these equations,and prove the following conclusions:let a?R#?R+,then a is a partial isometry if and only if one of the following equations has at least one solution in ?a xa=x(a+)*;aa*x-aa+x;a*ax=a-+ax;a*xa=a+xa and x=xaa*,where ?a={a,a*,a+,a#,(a#)*,(a+)*}.