The annulus as a K-spectral set (Q1951378)

For a fixed constant \(K \geq 1\), a closed subset \(X\) of the complex plane which contains the spectrum of a Hilbert space bounded linear operator \(A\) is said to be a \(K\)-spectral set for \(A\) if the inequality \(\|f(A\| \leq K \sup\{|f(z)| : z \in X\}\) holds for all bounded rational functions on \(X\). The author obtains several results concerning classes of Hilbert space operators having a given annulus as a \(K\)-spectral set. An extension of a result of \textit{J. G. Stampfli} [Pac. J. Math. 23, 601--612 (1967; Zbl 0152.33802)] concerning the numerical radius of an operator and of its inverse is also obtained.