Numerical radii of zero-one matrices (Q583574)

Stimulated by the work of V. Müller, the authors study the problem: What is the smallest constant C such that \[ w(TS)\leq Cw(T)\| S\| \] for all commuting bounded linear operators S and T on a complex Hilbert space? Here \(w(A)=\sup \{| (Ax,x)| /(x,x),\quad x\neq 0\}.\) They show that, if S is the shift on the Hilbert space of dimension 9 and \(T=S^ 3+S^ 7\), then \(w(T)=\cos (\pi /10)\) and \(w(TS)=1\). Thus \(C\geq 1/\cos (\pi /10)>1.05\). Also, if S is the shift on the Hilbert space of dimension 16 and \(T=S^ 4+S^{14}\), then \(w(T)=\cos (\pi /9)<1/1.064\) and \(w(TS)=1\). The authors also present some implications for dilation theory.