If A and B are bounded linear operators on an infinite dimensional complex Hilbert space H \mathcal {H} , let Ï ( X ) = A X â X B \tau (X)\, = \,AX\, - \,XB (X in L ( H ) \mathcal {L}(\mathcal {H}) ). It is proved that Ï ( Ï ) = Ï ( Ï | C p ) ( 1 â©œ p â©œ â ) \sigma (\tau )\, = \,\sigma (\tau |{C_p})\,(1\, \leqslant \,p\, \leqslant \infty ) , where, for 1 â©œ p > â 1\, \leqslant p\, > \,\infty , C p {C_p} is the Schatten p-ideal, and C â {C_\infty } is the ideal of all compact operators in L ( H ) \mathcal {L}(\mathcal {H}) . Analogues of this result for the parts of the spectrum are obtained and sufficient conditions are given for Ï \tau 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.