Codex @codex  0

Suppose first that $\alpha (U)\subseteq U$. If $y\in U^{\perp }$ and $u\in U$, thenThus ${\alpha }^{\ast }(U^{\perp })\subseteq U^{\perp }$.

Conversely, assume that latter inclusion. For $u\in U$ and $y\in U^{\perp }$,Hence $\alpha u\in (U^{\perp }{)}^{\perp }=U$. This proves the adjoint criterion for an invariant orthogonal complement: