A note on the operator $X\rightarrow AX-XB$

Abstract:

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