Inner derivations and norm equality (Q2758984)

The authors study properties of inner derivations restricted to normed ideals \(J\) in the algebra \(L({\mathfrak h})\) of bounded operators on a Hilbert space. Let \(\delta(A)X= AX- XA\), \(X\in L({\mathfrak h})\) be an inner derivation and denote by \(\delta(J,A)\) the restriction of \(\delta(A)\) to the normed ideal \(J\). Let \(N_0(A)\) be the number \(N_0(A)= 2\inf_{\lambda\in C}\|A-\lambda I\|\). \(A\) is said to be \(S\)-universal ifNEWLINENEWLINENEWLINE(i) \(\delta(J, A)= N_0(A)\) for all \(J\).NEWLINENEWLINENEWLINEIt is proved that \(A\) is \(S\)-universal if and only ifNEWLINENEWLINENEWLINE(ii) \(\text{diam }W(A)= N_0(A)\), \(W(A)= \{(Ax,x):\|x\|= 1\}\) and if and only ifNEWLINENEWLINENEWLINE(iii) \(\text{diam }\sigma(A)= N_0(A)\).NEWLINENEWLINENEWLINEThis theorem is employed to extend the theory of \(S\)-universal operators introduced by \textit{L. Fialkow} [Isr. J. Math. 32, 331-348 (1979; Zbl 0434.47006)]. The theorem utilizes a result which is of independent interest:NEWLINENEWLINENEWLINE(iv) \(\|A+ B\|=\|A\|+\|B\|\) if and only if \(\|A\|\|B\|\in W(A^* B)\).