Hong Ke, DUYU, E QING WANG
Acta Mathematica Sinica. 2009, 25(4): 0-0.
Using the technique of block-operators, in this note, we prove that
if $P$ and $Q$ are idempotents and $(P-Q)^{2n+1}$ is in the trace
class, then $\hbox{tr}(P-Q)^{2m+1}$ is also in the trace class and
$\hbox{tr}(P-Q)^{2m+1}=\dim({{\mathcal R}(P)}\cap{{\mathcal
R}(Q)}^\perp)-\dim({{\mathcal R}(P)}^\perp\cap{\mathcal R}(Q)),
\hbox{ for all } m\geq n.$ Moreover, we prove that $\dim({{\mathcal
R}(P)}\cap{{\mathcal R}(Q)}^\perp)=\dim({{\mathcal
R}(P)}^\perp\cap{\mathcal R}(Q))$ if and only if there exists a
unitary $U$ such that $UP=QU\hbox{ and }PU=UQ,$ where ${\mathcal
R}(T)$ denotes the range of $T.$
