Norm of a derivation and hyponormal operators (Q2766078)

Let \(({\mathcal J}, \|\cdot\|_{{\mathcal J}})\) be a two-sided norm ideal in the algebra \({\mathcal L}(H)\) of all bounded linear operators on a complex Hilbert space \(H\). For \(A\in{\mathcal L}(H)\), the inner derivation induced by \(A\) is the operator \(\delta_A\) defined on \({\mathcal L}(H)\) by \(\delta_A(X)= AX-XA\), \(X\in {\mathcal L}(H)\). The restriction \(\delta_{{\mathcal J},A}\) of \(\delta_A\) to \({\mathcal J}\) is a bounded linear operator on \({\mathcal J}\), and \(\|\delta_{{\mathcal J},A}\|\leq\|\delta_A\|=2d(A)\), where \(d(A)=\inf\{\|A-\lambda\|: \lambda\in{\mathbb C}\}\). An operator \(A\in{\mathcal L}(H)\) is called \(S\)-universal if \(\|\delta_{{\mathcal J},A}\|=2d(A)\) for each norm ideal \({\mathcal J}\) in \({\mathcal L}(H)\). \textit{L. A. Fialkow} [Isr. J. Math. 32, 331-348 (1979; Zbl 0434.47006)] showed that a subnormal operator is \(S\)-universal if and only if the diameter of the spectrum is equal to the diameter of the smallest disk containing it. Continuing this investigation, the authors prove that the same conclusion holds true for an arbitrary hyponormal operator.