Not all quadrative norms are strongly stable (Q1866439)

For any \(k\geq 1\), a norm \(N(.)\) on an algebra \({\mathcal A}\) over \(\mathbb{R}\) or \(\mathbb{C}\) is said to be \(k\)-bounded if \(N(x^k)\leq N(x)^k\) for all \(x\) in \({\mathcal A}\). In this paper, an example of a two-bounded (also called quadrative) but not three-bounded norm is given. More precisely, if \({\mathcal A}\) is the algebra consisting of matrices \(A=\alpha E+\beta E^2+\gamma E^3\), where \(E\) is the 4-by-4 Jordan block \[ \left( \begin{matrix} 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\end{matrix} \right), \] then the norm \(N(A)=u|\alpha|+ v|\beta |+w|\gamma|\), where \(u\), \(v\) and \(w\) satisfy \(0<v\leq u^2\) and \(0<u^3<w\leq uv+v\sqrt{u^2-v}\) is one such example. A slight modification even yields one on a unital algebra.