Question about operators on Hilbert space

Question

Let $\cal{H}$ be a Hilbert space, $P_1,P_2,\cdots,P_m$ a sequence of orthonormal projections such that $P_iP_j=0$ for $i\neq j$ and $P_1+P_2+\cdots+P_m=I$. Then $\|\sum^m_{k=1}P_kTP_k\|\leq\|T\|$ for all $T\in \cal{B}(H)$. Is it true? If it is true, how to prove?

Answer 1

For $x\in H$, we have $$ \begin{split} \|\sum_k P_kTP_kx\|^2 &= \langle \sum_j P_jTP_jx,\sum_k P_kTP_kx \rangle = \sum_{j,k} \langle TP_jx,P_jP_kTP_kx\rangle \\ &= \sum_{j} \langle P_jTP_jx,P_jTP_jx\rangle \leq \|T\|^2 \sum_{j} \langle P_jx,P_jx\rangle \\ & = \|T\|^2 \sum_{j} \langle P_jx,x\rangle = \|T\|^2 \langle \sum_{j} P_jx,x\rangle \\ &= \|T\|^2 \|x\|^2. \end{split} $$