A note on spectrally dominant norms
Let \(A\) be a finite-dimensional associative complex algebra. A vector space norm on it is called stable if there exists a positive constant \(\sigma\) such that \(\|x^n\|\leq\sigma\|x\|^n\) for every \(x\in A\) and all \(n\). It is known that stable norms on \(A\) are spectrally dominant (i.e., for every \(x\in A\), the spectral radius of \(x\leq\|x\|\)). A theorem of \textit{S. Friedland} and \textit{C. Zenger} [Linear Algebra Appl. 58, 97-107 (1984; Zbl 0543.15018)] asserts the converse. This theorem is not true for arbitrary \(A\), even if it is commutative and has a unit. The paper studies the limits of the theorem and shows that \(\text{Rad}(A)\) is a direct summand of \(A\) is sufficient to ensure that all spectrally dominant norms are stable. This condition becomes necessary if \(A\) is either commutative or has a unit. Also studied is a general version of the Friedland-Zenger theorem with a notion of almost stable norms.
scientific article; zbMATH DE number 1539379