Abstract: In this article we prove the following result. Let ▫$n ge 1$▫ be some fixed integer, and let ▫$R$▫ be a prime ring with ▫$2n le char(R) ne 2$▫. Suppose there exists an additive mapping ▫$T: R to R$▫ satisfying the relation ▫$T(x^{2n+1}) = sum_{i=1}^{2n+1}(-1)^{i+1} x^{i-1} T(x)x^{2n+1-i}$▫ for all ▫$x in R$▫. In this case, ▫$T$▫ is of the form ▫$4T(x) = qx + xq$▫ for all ▫$x in R$▫, where ▫$q$▫ is some fixed element from the symmetric Martindale ring of quotients. This result makes it possible to solve some functional equations in prime rings with involution which are related to bicircular projections. Keywords:bicircular projection, derivation, functional identity, Jordan tripple derivation, prime ring, ring with involution, semiprime ring Published: 07.06.2012; Views: 660; Downloads: 53 Link to full text

Abstract: It is shown that for many finite dimensional normed vector spaces ▫$V$▫ over ▫$Cc$▫, a linear projection ▫$P: V to V$▫ will have nice structure if ▫$P + lambda (I-P)$▫ is an isometry for some complex unit not equal to one. From these results, one can readily determine the structure of bicircular projections, i.e., those linear projections ▫$P$▫ such that ▫$P + mu (I - P)$▫ is an isometry for every complex unit ▫$mu$▫. The key ingredient in the proofs is the knowledge of the isometry group of the given norm. The proof techniques also apply to real vector spaces. In such cases, characterizations are given to linear projections ▫$P$▫ such that ▫$P - (I - P) = 2P - I$▫ is an isometry. Keywords:bicircular projection, symmetric norms, unitarily invariant norms, unitary congruence invariant norms, unitary similarity invariant norms Published: 10.07.2015; Views: 267; Downloads: 47 Link to full text