The norm of a derivation. (English) Zbl 0197.10501

Stampfli, Joseph Gail

Pac. J. Math. 33, 737-747 (1970).

Summary: In this paper, we determine the norm of the inner derivation \(\mathfrak Q_T\colon A \to TA - AT\) acting on the Banach algebra \(\mathfrak B(H)\) of all bounded linear operators on Hilbert space. More precisely, we show that \(\Vert\mathfrak Q_T\Vert = \inf\{2\,\Vert T - \lambad I\Vert : \lambda\] complex \(\}\). If \(T\) is normal, then \(\Vert\mathfrak Q_T\Vert can be specified in terms of the geometry of the spectrum of \(T\).

MSC:

47-XX	Operator theory
46-XX	Functional analysis

Keywords:

norm; inner derivation; Banach algebra; bounded linear operators; Hilbert space