If $X$ is a normed linear space and $S(X)$ its unit sphere, $X′$ its dual space and $Π=\{(x,f)∈S(X)×S(X′) \ | \ f(x)=1\}$, then for an operator $T$ on $X$, the numerical range $V(T)$ is defined by $V(T)=\{f(Tx) \ | \ (x,f)∈Π\}$. A projection on a complex Banach space $X$ is said to be hermitian if its numerical range is real.

I tried to find an example of an hermitian operator on the Banach space $C_{\mathbb C}[0,1]$ but could not find it. I have some intiutions that the decomposition $E\oplus F$ of $C_{\mathbb C}[0,1]$ where $E=\{f=\alpha \ | \ \mbox{for some } \alpha \in \mathbb C\}$ and $F=\{g=g_1+i g_2 \in C_{\mathbb C}[0,1] \ | \ \int g=\int g_1 + i \int g_2 =0\}$ is an hermitian decomposition, i.e., the projection $P: C_{\mathbb C}[0,1] \to E$ is hermitian, but could not prove it.

Does anyone know an example of an hermitian projection on $C_{\mathbb C}[0,1]$?

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