Numerical radii of simple powers (Q1369350)

The author considers the following problem due to John Holbrook [cf. \textit{K. R. Davidson} and \textit{J. A. R. Holbrook}, Mich. Math. J. 35, NO. 2, 261-267 (1988; Zbl 0692.47005)]: If \(S\) is an operator on a complex Hilbert space with operator norm \(|S|\leq 1\), \(\omega(T)\) denotes the numerical radius of the operator \(T\), is it true that for every positive integer \(n\), \(\omega(S^{n+1}) \leq\omega (S^n)\)? A counterexample to \(\omega (S^4) \leq\omega (S^3)\) with \(|S|=1\) is given in this paper.