On the “zero-two” law for positive contractions

Abstract

Let (X, Σ,μ) be a measure space (where μ is a positive σ-additive measure) and let Lp(X,Σ,μ), 1≦p≦ + ∞ be the usual real Banach lattices.

On dominant contractions and a generalization of the zero–two law

“…Using certain properties of L 1 -spaces Zaharopol [10] by means of the following theorem reproved Theorem 1.1. Theorem 1.2.…”

A Note on Dominant Contractions of Jordan Algebras

“…(ii) Unfortunately, Theorem 3.3 is not longer true if one replaces L 1space by an L p -space, 1 < p < ∞. The corresponding example was provided in [17]. (iii) It would be better to note that certain ergodic properties of dominant positive operators has been studied in [8] in a noncommutative setting.…”

“…A linear operator T : L 1 (X, F , µ) → L 1 (X, F , µ) is called positive contraction if T f ≥ 0 whenever f ≥ 0 and T ≤ 1. In [17] the following theorem was proved.…”

On a generalized uniform zero-two law for positive contractions of noncommutative $L_{1}$ -spaces and its vector-valued extension

“…Interchanging "sup" and "lim" in the strong zero-two law we have the following uniform zero-two law, proved by Foguel [12] using ideas of [38] and [11]. Zahoropol [44] has provided another proof of Theorem 1.2 which is given in the following theorem. Theorem 1.3.…”

“…Theorem 1.4. [44] Let T, S : L 1 → L 1 be two positive contractions such that T ≤ S. If S − T < 1 then S n − T n < 1 for all n ∈ N.…”