Adjoint Operator and Subspaces of Hilbert Spaces

Question

Let $H_1$ , $H_2$ Hilbert Spaces with $T:H_1 \to H_2$ adjoint operator and $M_1 < H_1$ , $M_2 < H_2$ their subspaces. Show that:

$T(M_1) \subset M_2 $ if, and only if, $T^*({M_2}^{\perp}) \subset {M_1}^{\perp} $

Answer 1

\begin{align} TM_1 \subseteq M_2 &\iff TM_1 \perp M_2^\perp \\ &\iff M_1 \perp T^*M_2^\perp \\ &\iff T^*M_2^\perp \subseteq M_1^\perp \end{align}