Ein starkes ``Null-Zwei''-Gesetz in \(L^ p\). (A strong ``zero-two'' law in \(L^ p)\)
Let T be a positive linear contraction on the space \(L^ p=L^ p(\mu)\) (p\(\geq 1\) is fixed) with \(\sigma\)-finite measure \(\mu\). The following two theorems are the main results of the presented paper. Theorem 1.2. The inequality \(\inf_{n\in {\mathbb{N}}}\| | T^ n- T^{n+1}| \|_{L^ p}<2\) implies that \(\lim_{n\to \infty}\| | T^ n-T^{n+1}| \|_{L^ p}=0.\) Theorem 2.2 If there exist \(\epsilon >0\) such that \[ \inf_{n\in {\mathbb{N}}}\| | T^ n-T^{n+1}| f\|_{L^ p}\leq 2-\epsilon \quad (f\in L^ p) \] then \[ \lim_{n\to \infty}\| | T^ n- T^{n+1}| f\|_{L^ p}=0\quad (f\in L^ p). \]
scientific article; zbMATH DE number 4044463