The numerical range of positive operators on Banach lattices (Q496796)

Let \(A\) be an operator in a Banach space \(X\). The spatial numerical range of \(A\), which is an extension of the concept of numerical range of operators on a Hilbert space, is defined as \[ V(A)=\{\varphi(Ax): x\in X,\;\varphi\in X',\;\|\varphi\|=\|x\|=\varphi(x)=1\}, \] and let \(v(A)=\sup\{|\lambda|: \lambda\in V(A)\}\). For a positive operator \(A\) in a Banach lattice, the authors consider the following four main questions: {\parindent=6mm \begin{itemize} \item[(1)] Is \(v(A)\in V(A)\)? \item [(2)] For a regular operator \(B\) with \(|B|\leq A\), is \(v(B)\leq v(B)\)? \item [(3)] Is \(V(A)\) symmetric with respect to the real axis? \item [(4)] For an irreducible element \(A\) with \(e^{i\theta}v(A)\in V(A)\), is \(V(A)=e^{i\theta}V(A)\)? \end{itemize}} The authors investigate these questions and answer them for both the numerical radius and the block numerical radius.