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Let $H$ be a separable Hilbert space and $L$ be a subspace. Let $B_i$ where $i$ ranges from 1 to n are bounded operators on $H$ and $P_L$ be a projector onto $L$.

If there is a relation $P_LB_iB_jP_L=b_{ij}P_L$ for some coefficients $b_{ij}$, then how can I show that the matrix $[b_{ij}]$ is a Hermitian positive? Could anyone please help me?

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False: take $n=1,L=H=\mathbb C^{2}$ and let $B=B_1$ be given by the matrix with rows $(-i,0),(0,-i)$. Then $P_L=I$ and $P_LBBP_L=(-1)I$.

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