In reading a paper on numerical quadrature I've come across a result that is proved in a manner that is very clever:

Let $X \subset \mathbb{C}$ be a compact, convex set. If $U$ is a finite-dimensional subspace of $L^2(X)$, $P_U: L^2(X) \rightarrow U$ is the orthogonal projection onto $U$, and $T_U: U \rightarrow L^2(X)$ is the multiplication operator $\varphi(z) \mapsto z\varphi(z)$ restricted to $U$. Then the spectrum of $P_U \circ T_U$ is contained in $X$.

The proof is quite clever. Notice that $P_U \circ T_U$ is a map $U \rightarrow U$; it has only point spectrum. Suppose $P_UT_U\varphi = \lambda \varphi$ with $\lambda \notin X$. Then using the geometric assumptions on $X$, we can always find a $\mu \notin X$ such that $|\lambda - \mu| > |x - \mu|$ for all $x \in X$. But then $P_U T_U \varphi - \mu \varphi = (\lambda - \mu) \varphi$, from which we can deduce that there is an $x_0 \in X$ such that $|x_0 - \mu| > | \lambda - \mu|$, a contradiction.

This is all well and good, but I'm interested in knowing whether there is a more functional-analytic proof for this result, something that does not involve the trick with $\mu$. The primary motivation for this is to attempt to extend some of the results in the paper. One of the most basic results in the theory is that any multiplication operator $M_f: L^2(X) \rightarrow L^2(X)$ has spectrum equal to the essential range of $f$, if $f \in L^{\infty}(X)$. I have been trying to use this fact, since $T_U$ is just $M_z$ restricted to $U$, but I haven't been successful so far. Am I doomed to have to use this geometric trick, or is there another approach?

Spectra of Multiplication Operators as a Numerical Toolby Vioreanu and Rokhlin. – Christopher A. Wong Oct 24 '12 at 18:14