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



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> 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 $X$.
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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?



Finding the spectrum of the composition of a projection with a multiplication operator
I would like to record an alternative method for proving the claim. First, there is a corollary of the Toeplitz-Hausdorff theorem:



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> Let $A$ be a bounded normal operator on a Hilbert space. Then the closure of its numerical range, $W(A)$, given by all values $\langle Ax, x \rangle$ with $\|x\|= 1$, is equal to the convex hull of $\sigma(A)$.
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Now, using the definitions of $P\_U$ and $T\_U$ from my question, notice that if $\lambda \in P\_U \circ T\_U$, then for the eigenfunction $\varphi$, $P\_U \varphi = \varphi$ since $T\_U$ is the multiplication operator restricted to $U$, so $P\_U T\_U \varphi = \lambda \varphi$, from which we obtain $\langle T\_U \varphi, \varphi \rangle = \lambda \|\varphi\|^2$, hence $\lambda \in W(M\_z)$, where $M\_z$ is the multiplication operator by $z$. $M\_z$ is a normal operator, because $z$ and $\bar{z}$ commute; thus the corollary from above yields


$$ \lambda \in \sigma(P\_U \circ T\_U) \Longrightarrow \lambda \in \mathrm{ch}(\sigma(M\_z)) = \mathrm{ch}(X)$$


Thus, if $X$ is itself convex, this implies that $\lambda \in X$.


While also simple, this approach is of course significantly less elementary than the method of proof used in the paper, but nevertheless perhaps somebody else besides myself may find it of interest.


