Restricted unitary group ($U_{\mathrm{res}}(H,P)=\{u:[P,u]\in {\mathcal{S}}_2\}$)

For an orthogonal projection $P$ on a complex Hilbert space, this is the group of unitary operators whose off-diagonal blocks relative to $PH\oplus (I-P)H$ are Hilbert-Schmidt operators. Its usual topology combines strong operator convergence with Hilbert-Schmidt convergence of $[P,u]$. It is the group implementable in the corresponding polarized fermionic Fock space.