If A and B are bounded linear operators on an infinite dimensional complex Hubert space %, let t(X) = AX -XB (X in £(0C)). It is proved that <j(t) = <j(t|C¡,) (1 < p < oo), where, for 1 < p < oo, Cp is the Schatten /»-ideal, and Cx is the ideal of all compact operators in £(9C). Analogues of this result for the parts of the spectrum are obtained and sufficient conditions are given for t to be injective. It is also proved that if A and B are quasisimilar, then the right essential spectrum of A intersects the left essential spectrum of B.