Analogues of the ``zero-two`` law for positive linear contractions in L p and C(X)
Let a \(\sigma\)-finite measure space be given, and choose a positive linear contraction (a nonexpansive mapping) T, defined on the related L p space \((1\leq p<\infty)\). The zero-two law is the statement that for \(p=1\) \[ (*)\quad \sup (\lim_{n\to \infty}\| T\quad nf-T^{n+1}f\|_ 1)\quad is\quad either\quad zero\quad or\quad two, \] the sup being taken over all f's in the unit ball of L 1, a result proved by Ornstein and Sucheston. In this paper, the author proves many variants of this theorem, including (i) if \(\alpha_ p\) is a `natural' constant which is 2 when \(p=1\), and is the best possible choice, then \[ \sup (\lim_{n\to \infty}\| T\quad nf-T^{n+1}f\|_ p\quad is\quad either\quad zero\quad or\quad \geq \alpha_ p, \] the sup being taken over the unit ball of L p; moreover, (ii) \[ \sup (\lim_{n\to \infty}\| T\quad nf- T^{n+1}f\|_ p\quad is\quad either\quad 2^{1/p}\quad or\quad zero, \] if the sup is taken over nonnegative f's in the unit ball of L p. There are also extensions concerning the uniform case of (*) (when sup and lim are interchanged, thus yielding operator norms), and the case of sup-norms (a.e. bounded or continuous functions); finally, some improvements are announced (1.10) and open questions raised (2.2).
scientific article; zbMATH DE number 4067613