Let $M$ and $N$ be closed subspaces of a Hilbert space $H$ and let $P_M$ and $P_N$ denote the orthogonal projections onto $M$ and $N$, respectively. I have proved so far the following equivalences:
(a) $P_M^\perp|N$ and $P_M|N^\perp$ are compact.
(b) $P_M^\perp P_N$ and $P_M P_N^\perp$ are compact (as operators in $H$)
(c) $P_M - P_N$ is compact.
I would like to know what this actually means for the subspaces $M$ and $N$ (not in terms of orthogonal projections). Does anyone know anything about this?