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Let $T$ be a Hilbert space operator. Its numerical range is \begin{equation} W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.\end{equation}

It is a well-known fact that $W(T)$ is a convex subset of the complex plane. However, every proof I know is by brute force computation. First for $2\times 2$ matrices, then the general case.

Even though the computation can be carried out in clever ways, it still fails to provide some explanation why this is true. What is the link between this result and other concepts of the theory?

I wonder whether there is any conceptual explanation for this result. I do not ask the explanation to be rigorous, just some ideas.

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    Fejér's theorem that inspired Toeplitz is: Let $f$ and $g$ be two $2\pi$-periodic continuous and real-valued functions. Consider the convex hull $C$ of the curve $t \mapsto (f(t),g(t))$ in the real plane. Then for each $n$ the curve $t \mapsto (\sigma_n(f,t), \sigma_n(g,t))$ given by the $n$th [Fejér sums](http://www.encyclopediaofmath.org/index.php/Fejer_sum) of $f$ and $g$ (the arithmetic means of the first $n+1$ Fourier approximations of $f$ and $g$) lies entirely in $C$. Toeplitz recast this result first in terms of Laurent series (his thesis) and then in terms of infinite binary forms.2012-07-17

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