Transformations satisfying homogeneous identities (Q584325)

The author announces a program of investigation of linear maps which satisfy homogeneous identities on commutative algebras. The present paper may be considered as a first step. In essence its main result reads as follows. Suppose that there are given a commutative algebra A over a field K, a linear map \(0\neq T\in Hom_ K(A,A)\) and \(c_ 0,c_ 1,...,c_ 4\in K\), not all zero, such that (*) \(c_ 0xy+c_ 1xT(y)+c_ 2yT(x)+c_ 3T(xy)+c_ 4T(x)T(y)=0\) holds for all \(x,y\in A\). Then under some mild additional assumptions either \(T=dI_ A+D\) for some derivation D of A and \(d\in K\), or \(T=eI_ A+fE\) for some algebra endomorphism E of A and \(e,f\in K\). (As for the converse it is easily verified that linear maps of this type satisfy identities like (*).) Several sets of assumptions are discussed under which this result is valid as well as two instructive examples are given.