A note on Gâteaux differentials of hermitian elements in Banach algebras

Herzog, Gerd
Schmoeger, Christoph

facul_08; 07

Date: 07. 03. 2008
Abstract:
Let $\cA$ be a complex unital Banach algebra with unit {\bf 1}. If $a \in \cA$ is hermitian then we show that
\bn
\|a\|^2 = \lim_{t \to 0+} t^{-1} (\| {\bf 1} + ta^2 \|-1) \, ,
\en
and we give a proof of an inequality due to J. Nieto.

Primary MSC: 47A12 Numerical range, numerical radius
Keywords: numerical range, hermitian element