A remark on the uniform zero-two law for positive contractions
The author considers many recent zero-two law results for positive contractions on Banach lattices, i.e. results stating that if T is a positive contraction on a Banach lattice, then \(\| T^ n- T^{n+1}\| =2\) for all \(n\in {\mathbb{N}}\) or \(\| T^ n-T^{n+1}\| \to 0\) as \(n\to \infty\). Then he goes to show that this zero-two law holds for positive contractions on arbitrary Dedekind complete Banach lattices, this way improving some of the previous results. Furthermore, he is able to characterize T's such that \(\| T^ n(I-T^ k)\| \to 0\) as \(N\to \infty\) as exactly those T's for which the peripheral order spectrum contains no \(\lambda\) with \(| \lambda | =1\) and \(\lambda^ k\neq 1\). As a consequence one has that \(\| T^ n(I-T^ k)\| =2\) for all \(n\geq 0\), \(k\geq 1\) if and only if the peripheral order spectrum of T equals \(\{z:| z| =1\}\).
zero-two law results for positive contractions on Banach lattices
scientific article; zbMATH DE number 4151166
scientific article