A bounded operator $T$ on a Hilbert space is called quasitriangular if there exists an increasing sequence of finite rank projections $P_n$ converging pointwise to the identity such that $||TP_n-P_nTP_n||\to 0$.
I saw mentioned in several places that compact operators are quasitriangular. Is this something obvious? Can someone please show me a proof, or give me a reference for this fact?
Being compact, $T$ is a limit of finite-rank operators, say $T=\lim T_m$. Define $P_n$ to be the range projection onto the union of the ranges of $T_1,\ldots,T_n$. So $P_nT_m=T_m$ if $m\leq n$.
Now fix $\varepsilon>0$. Then there exists $m$ such that $\|T-T_m\|<\varepsilon/2$. For any $n\geq m$, we have $$ \|P_nT-T\|\leq \|P_nT-P_nT_m\| + \| P_nT_m-T_m\|+\|T_m-T\|<\frac\varepsilon2+0+\frac\varepsilon2=\varepsilon, $$ So $P_nT\to T$. Taking adjoints (and using that the adjoint of a compact is compact), $TP_n\to T$. Thus $$ \|P_nT-P_nTP_n\|\leq \|T-TP_n\|\to0. $$