Commutator and self-commutator approximants. II (Q2853184)

The present author in [Proc. Am. Math. Soc. 115, No. 4, 995--1000 (1992; Zbl 0773.47020)] and [Proc. Am. Math. Soc. 134, No. 1, 157--165 (2006; Zbl 1072.47013)] as well as \textit{S. Bouali} and \textit{S. Cherki} in [Acta Sci. Math. 63, No. 1--2, 273--278 (1997; Zbl 0892.47035)] approximated in the von Neumann-Schatten norm \(\| \cdot\| _p\) (\(1\leq p<\infty\)) a Hilbert space operator by a commutator \(AX-XA\), by a self-commutator \(X^*X-XX^*\), and by a generalized commutator \(AX-XB\), respectively. Here, the author investigates the approximation in the sup norm. To this end, he shows that the power norm equality \(\| A^n\| =\| A\| ^n\) holds for any positive integer \(n\) and any paranormal operator \(A\).