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

Abstract

Abstract. Zaharopol proved the following result: let T, S : L 1 (X, F , µ) → L 1 (X, F , µ) be two positive contractions such that T ≤ S. If S − T < 1 then S n − T n < 1 for all n ∈ N. In the present paper we generalize this result to multi-parameter contractions acting on L 1 . As an application of that result we prove a generalization of the "zero-two" law.

ON A MULTI-PARAMETRIC GENERALIZATION OF THE UNIFORM ZERO-TWO LAW IN L1-SPACES

“…The proved theorem is a multi-parametric generalization of the main result of [12]. Hence, it generalizes all main results of [3,4,6,13,20].…”

“…In all these investigations, the generalization was in direction replacement of the L 1 -space by an abstract Banach lattice (see [8,11,15,16,18]). In [12] we have proposed another kind of generalization of the uniform zero-two law in L 1 -spaces. In this paper we continue the previous investigations and prove a multiparametric generalization of the uniform "zero-two" law in L 1 -space.…”

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

“…The aim of this paper is to prove a non-commutative version of a generalized uniform "zero-two" law for multi-parametric family of positive contractions of L 1 -spaces associated with von Neumann algebras. As a particular case (when the algebra is commutative), we recover the results of [30,31]. Moreover, we emphasize that Theorem 1.2 will be included in the main result as a particular case.…”

“…In [23,34,30] we have extended the last result for several kind of Banach spaces. Therefore, it is natural step is to find analogous of Theorem 1.3 in a non-commutative setting.…”

The strong ``zero-two" law for positive contractions of Banach--Kantorovich $L_p$-lattices

“…for almost all ω ∈ Ω . Due to [20,Theorem 2.1] we then obtain that (S 1 ) ω (S 2 ) n ω − (T 1 ) ω (T 2 ) n ω 1,ω < 1 for every n ≥ n 0 and for almost all ω ∈ Ω. Hence, S 1 S n 2 − T 1 T n 2 (ω) = (S 1 ) ω (S 2 ) n ω − (T 1 ) ω (T 2 ) n ω 1,ω < 1 for every n ≥ n 0 and for almost all ω ∈ Ω.…”