# Difference of Orthogonal Projections is Compact

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?

• Thanks, but why are U and V orthogonal to each other? – Friedrich Philipp Mar 15 '17 at 21:19
• Right. But then $(P_M-P_N)|_U\neq I_U$ in general. – Friedrich Philipp Mar 16 '17 at 21:04