Two Types Of Preserver Problems Related To Idempotent Matrix

The purpose of preserver problems is characterizing the maps between two algebra-ic structures, where some algebraic properties maintain unchanged through these maps’ function. Research on preserver problems starts from an article published in1897by Frobenius, in which linear maps preserving determinant on spaces of square matrices over complex field are characterized. By now, there are already plenty of results on pre-server problems on algebras of matrices and algebras of operators. Invariant involved in preserver problems can be divided into functions, subsets, relations and transformation, while preservers can be linear maps, multiplicative maps, additive maps, etc.In2003, maps preserving some properties of A-AB were studied for first time by Semrl, and continuous bijective maps preserving idempotent property of A-κB on matrix spaces over complex field were characterized. From then on, many scholars have done a lot of work on related preserver problems that improved former’s results and enriched the content of preserver problems. Maps preserving k-potent property of A-AB in two directions on square matrix spaces over complex field was characterized in2007by You et al., and maps preserving k-potent property of A-AB in two directions on triangular matrix spaces over complex field was characterized in2008by You et al. A preserver in two directions must be a preserver in one direction, but the reverse is not true. Cao et al. improved the results by You et al., and characterized maps preserving k-potent property of A-AB in one direction on square matrix spaces over complex field.In this paper, maps preserving preserving k-potent property of A-AB in one direc-tion on symmetric matrix space over complex field were characterized, which included the forms of preservers in one direction. Further, it is not only the result improved the for-mers’and the method used in research is suitable for maps preserving k-potent property of A-AB in one direction on square matrix spaces over complex field.One of motives for development of preserver problems is researching on new invari-ant. In2010, Pazzis presented the conditions in which a matrix over an arbitrary field can be expressed as linear combination of two idempotent matrices, and based on these conditions, he proved that every matrix over an arbitrary field can be expressed as linear combination of three idempotent matrices. In2012, Botha presented the sufficient and necessary conditions in which a matrix over an arbitrary field can be decomposed as sum of two nilpotent matrices. Considering preserver problems, it is an attractive problem to study the maps preserving these types of matrix decomposition. And it is necessary to do further research on these two types of matrix decomposition before studying preserver problems related to them.Although Pazzis and Botha presented the conditions in which matrix can be decom-posed as these two forms, they did not pointed out the relation between these two types of matrix decomposition. In this paper, it is proved that decomposing a matrix as sum of two nilpotent matrices over a field is a special case of decomposing a matrix as linear combi-nation of two idempotent matrices over a field. In further, the conditions are presented in which a2x2matrix can be decomposed as sum of two nilpotent matrices over the integer ring, meanwhile, some special examples are discussed.