Let $H$ be a separable Hilbert space and let $T:H \rightarrow H$ be a symmetric bound linear map.
a) Show that for every orthogonal projection $P$ on $H$ ($P' = P$, $P^2 = P$) PTP is symmetric.
b) Prove the existens of a sequence $C_n$ of compact symmetric linear maps such that $C_nx\rightarrow Tx$ for $x\in H$.
My try:
a) I don't see that $P^* = P$ for all orthogonal projection, why is it so? but if that is true $(PTP)' = P'T'P' = PTP$
b) I do not really know how to start here, I know that the limit of compact maps are compact if they converge uniformly. Any help or hint would be greatfull
Here's one way to see (a): write $\langle Px, y \rangle = \langle Px, Py \rangle + \langle Px, y-Py \rangle$. Since $P$ is orthogonal projection, $y - Py$ is orthogonal to the range of $P$, so the second term vanishes. Thus $\langle Px, y \rangle = \langle Px, Py \rangle$. By the same argument, $\langle x, Py \rangle = \langle Px, Py \rangle$. Since $\langle Px, y \rangle = \langle x, Py \rangle$, $P$ is symmetric.
For (b): fix an orthonormal basis $\{e_i\}$ for $H$, and let $P_n$ be orthogonal projection onto the span of $e_1, \dots, e_n$. Show that for any $x$, $P_n x \to x$. Now try taking $C_n = P_n T P_n$. (Note that $P_n \to I$ pointwise, but not uniformly, so this doesn't contradict the fact about compact operators that you state.)