Spatial numerical range of operators on weighted Hardy spaces (Q539403)

For any bounded linear operator \(T\) on a complex normed space \(X\), the spatial numerical range \(V(T)\) of \(T\) is defined as \(V(T)= \{ x^*(Tx): x\in X\), \(x^*\in X^*\) and \(\| x\|=\| x^*\|= x^*(x)= 1\}\). It is known that \(V(T)\) is always pathwise connected but is, in general, not convex. For \(1<p<\infty\), the weighted Hardy space \(H^p(\beta)\) with weights \(\{\beta(n)\}^\infty_{n=0}\), where \(\beta(n)> 0\) for all \(n\) and \(\beta(0)= 1\), is the reflexive Banach space consisting of formal power series \[ f(z)= \sum^\infty_{n=0}\widehat f(n) z^n \] with \[ \| f\|^p_p\equiv \sum^\infty_{n=0}|\widehat f(n)|^p \beta(n)^p< \infty. \] In this paper, the authors (1) give an example of a rank-2 operator \(T\) on \(H^p(\beta)\) such that \(V(T)\) is star-shaped with respect to \(0\) (\(tz\) in \(V(T)\) for all \(t\), \(0\leq t\leq 1\), and all \(z\) in \(V(T)\)), but not convex, and hence \(T- a\) is not star-shaped (w.r.t. \(0\)) for some \(a\) in \(V(T)\), and (2) prove that if \(T\) is a compact operator on \(H^p(\beta)\) with \(V(T)\) star-shaped (w.r.t. \(0\)), then \(V(T)\) is closed.