The Numerical Range of a Simple Compression

Abstract

The numerical range of the contraction
$K:\scriptsize \begin{bmatrix} a & b \cr c& d \end{bmatrix} \mapsto \scriptsize \begin{bmatrix} a & 0 \cr 0& 0 \end{bmatrix} $
acting on $L(\bC^2)$ is identified,
so allowing one to exhibit a hermitian projection that is not ultrahermitian.


An explicit formula for the norm of the operator
$\KM :=
\scriptsize
\begin{bmatrix}
a & b \cr c& d
\end{bmatrix}
\mapsto
\begin{bmatrix}
ma & b \cr c& d
\end{bmatrix}$ ($m \in \bC).$
translates into a family of inequalities in four complex variables.