Derivation and Jordan operators (original) (raw)

VX e s fA(X) = AX - XA; we denote R(~A), R(~A)- and {A}' respectively the range, the norm closure of the range and the kernel of ~A. We denote Af = {A e/:(g) : R(~A)- (2 {A*}' = {0}}. If H is finite-dimensional, Af = s If H is infinite-dimensional, this equality does not hold. So a reasonable purpose is to determine what elements are in N. When H is a separable Hilbert space, AY contains the operators A for which p(A) is normal for some quadratic polynomial p(z) [2 ],the subnormal operators with cyclic vectors [2 ] and the ison:letries [3 ]. In this paper, we show that AY contains also all the operators unitarity equivalent to Jordan operators.