A projection on a complex Banach space $X$ is said to hermitian if its numerical range is real.

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

  • $\begingroup$ Is the numerical range of an operator $\frac{\int \bar f A f}{\int \bar f f}$? $\endgroup$
    – s.harp
    Jan 13, 2016 at 10:07
  • $\begingroup$ No, it isn't. I follow the definition in the Duncan and Bonsall's book. But I am not sure whether the definitions coincide. $\endgroup$
    – cejvan
    Jan 13, 2016 at 10:51
  • $\begingroup$ Can you give the definition? $\endgroup$
    – s.harp
    Jan 13, 2016 at 11:27
  • $\begingroup$ If $X$ is a normed linear space and $S(X)$ its unit sphere, $X'$ its dual space and $\Pi= \{(x,f)\in S(X)\times S(X') \ | \ f(x)=1 \},$ then for an operator $T$ on $X$, the numerical range $V(T)$ is defined by $V(T)=\{f(Tx) \ | \ (x,f)\in \Pi\}.$ $\endgroup$
    – cejvan
    Jan 13, 2016 at 11:42


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