[Math] Range and kernel of linear operators

Question

I have a compact linear operator $T$, and I would like to show $$\operatorname{range}(\lambda I-T)=(\ker(\overline{\lambda}I-T^*))^\perp.$$
I have shown the forward inclusion "$\subset$" directly by using the definition of adjoint. However, I'm having trouble with the reverse inclusion "$\supset$"… it seems much harder to begin with the orthogonal complement with a kernel as well as to show that something is in a range. Any suggestions/hints would be appreciated. Thanks!

Answer 1

Note that you are missing the closure on your left-hand side.

Suppose that $x\in\text{range}(S)^\perp$. For every $y$, $$ 0=\langle x,Sy\rangle=\langle S^*x,y\rangle. $$ As $y$ is arbitray, $S^*x=0$. That is, $x\in\ker S^*$. So $$ \text{range}(S)^\perp\subset \ker S^*. $$ Taking orthogonals you get your inclusion.