Let $H$ be a complex Hilbert space, $T\in\mathfrak L(H)$ and $U,V$ be subspaces of $H$. Let $W:=U\oplus V$ denote the orthogonal direct sum of $U$ and $V$. Suppose we know $TU^\perp\subseteq U^\perp$ and $TV^\perp\subseteq V^\perp$.
Are we able to conclude $TW^\perp\subseteq W^\perp$? Maybe this is a consequence of $W^\perp=W^\perp\oplus V^\perp$. Can we show this?
I've tried to show $W^\perp=W^\perp\oplus V^\perp$, but neither of the inclusions is obvious to me. For example, if $x\in U^\perp\oplus V^\perp$, then there are $(y,z)\in U^\perp\times V^\perp$ with $y\perp z$ and $x=y+z$. Analogously, if $w\in U\oplus V$, there are $(u,v)\in U\times V$ with $u\perp v$ and $w=u+v$. We need to show that $\langle x,w\rangle_H=0$. Now, clearly, $\langle x,w\rangle_H=\langle y,v\rangle_H+\langle z,u\rangle_H$ ... Now I guess, if the claim holds, we somehow need to "add a $0$" in a clever way.