1
$\begingroup$

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

  • 0
    For (a): Didn't you write in your assumption on $P$ that you want to have $P' = P$? Where is your point? For (b): Let $(e_n)_{n\in\mathbb N}$ an orthogonal basis for $H$. Such a thing exists, as $H$ is seperable. Now let $P_n$ denote the orthogonal projection onto $\operatorname{span} \{e_1, \ldots, e_n\}$ and $C_n := P_nTP_n$. Then $C_n$ is symmetric by (a) and compact by finite-dimensionality.2012-12-10

1 Answers 1