Critical exponents and their generalizations (Q2707726)

Recall that the critical exponent of a normed linear space \(X\) is the smallest integer \(q\) such that if \(\|A\|\leq 1\) and \(\|A^q\|<1\), then the spectral radius \(r(A)<1\), for any linear operator \(A\) on~\(X\), see \textit{V. Pták} [Proc. Colloq. ``Convexity'', Copenhagen, 244-248 (1967; Zbl 0148.36904)]. In~the present paper, the author surveys recent results about the (non-)existence of the critical exponent. For instance, the critical exponent exists if the unit sphere of~\(X\) can be covered by finitely many analytic sets lying entirely outside the open unit ball. On~the other hand, the critical exponent does not exist if there is a semigroup \(A(t)\), \(t\geq 0\), of contractions and a point \(x_0\) such that \(\|A(t)x_0\|=1\) for some, but not all~\(t\). If~the unit sphere is piecewise analytic, a sort of converse to the last result also holds.NEWLINENEWLINEFor the entire collection see [Zbl 0933.00055].