Norms possessing a critical exponent (Q1112146)

The critical exponent of a norm in \(R^ n\) is defined as the minimal natural number q such that for all linear operators satisfying \(\| A\| =1\), the equality \(\| A^ q\| =1\) implies \(\| A^ m\| =1\) for \(m>q\). The existence and nonexistence of the critical exponent is considered from the point of view of the theory of analytic functions of several complex variables. Typical is the following Theorem 1. Let U be some bounded complex neighborhood of the unit ball \(D=\{v\in R^ n:\quad \| v\| \leq 1\}.\) If the sphere \(\{v\in R^ n:\quad \| v\| =1\}\) is contained in the set of the zeros of a function f, holomorphic in U and such that f(0)\(\neq 0\), then the critical exponent exists. The author notices that if one exploits the ``Noether property'' of the ring of germs of holomorphic functions in a ``dual'' way in comparison with the use made in the proof of the above Theorem 1, one can obtain an affirmative answer to a conjecture of Yu. V. Lyubich: Theorem 2. If the sphere is an analytic submanifold in \(R^ n\), then the critical exponent exists. Another important result is contained in Theorem 6. The \(l_ p\)-norm possesses a critical exponent for each p, \(1\leq p\leq \infty\).