The equation dd ′ + d ′ d = D 2 for derivations on C-algebras

Abstract

Let A be an algebra. A linear mapping d : A!A is called a derivation if d ( ab ) = d ( a ) b + ad ( b ) for each a;b 2A . Given two derivations d and d ′ on a C -algebra A , we prove that there exists a derivation D on A such that dd ′ + d ′ d = D 2 if and only if d and d ′ are linearly dependent.