A Few Remarks on ”Zero-Two” Law for Positive contractions in the Orlicz-Kantorovich spaces

In this paper we establish a vector version the ”zero-two” law for positive contractions in the Orlicz-Kantorovich spaces.


Introduction
Let (X, F, µ) be a measure space with a positive σ-additive measure µ. In what follows, for the sake of shortness, we denote by L 1 the usual L 1 (X, F, µ) space associated with (X, F, µ). A linear operator T : L 1 → L 1 is called a positive contraction if T f ≥ 0 whenever f ≥ 0 and ∥T ∥ ≤ 1. Ornstein and Sucheston [29] obtained an analytic proof of the Jamison-Orey result [20], and in their work they proved the following theorem [ (1. 1) or This result was later called a strong zero-two law. Consequently, [29,Theorem 1.3], if T is ergodic with T * 1 = 1 (e.g. T is ergodic and conservative), then either (1.1) holds, or ∥T n g∥ 1 → 0 for every g ∈ L 1 with ∫ g dµ = 0. Some extensions of the strong zero-two law can be found in [31,34].
Interchanging "sup" and "lim" in the strong zero-two law we have the following uniform zero-two law, proved by Foguel [8] using ideas of [29] and [7].
In [32] it was proved the following uniform strong "zero-two" law for positive contractions of L p -spaces (1 ≤ p < +∞): A "zero-two" law for Markov processes was proved in [3], which allowed to study random walks on locally compact groups. Other extensions and generalizations of the formulated law have been investigated by many authors [7,9,31,24]. In all these investigations, the generalization was in direction replacement of the L 1 -space by an abstract Banach lattice (see [25,32]). In [28] we have proposed another kind of generalization of the uniform zero-two law in L 1 -spaces.
Furthermore, in [13] we have established a vector-valued analog of uniform "zero-two" law for the positive contractions of the Banach -Kantorovich L p -lattices.
Later in [17] we have prove an vector-valued analog of Theorem 1.3.
In [21] it was proved the following "zero-two" law for positive contractions in the Banach lattices having a weak unit and uniformly monotone norm. Theorem 1.6. Let E be a Banach lattice having a weak unit and uniformly monotone norm.
Uniform monotone properties of Orlicz spaces considered in [1] and [2]. The monotonicity properties of the Luxemburg norm in Musielak -Orlicz spaces are characterized in [5]. In particular, if the Orlicz function M has △ 2 -condition, then Orlicz space is vector lattice with the uniform monotone norm and having a weak unit. Hence, if the Orlicz function M has △ 2condition, the uniform "zero-two" law is valid for positive contractions in the Orlicz spaces.
The main aim of this paper is to prove the uniform "zero-two" law for the positive contractions of the Orlicz -Kantorovich lattices L M (∇, µ). We notice that in [6,10,11], [14]- [16] several ergodic theorems have been obtained for positive contractions of L p (∇, µ)-spaces.

Preliminaries
Let (Ω, Σ, λ) be a measurable space with finite measure λ, and L 0 (Ω) be the algebra of all measurable functions on Ω ( here the functions equal a.e. are identified) and let ∇(Ω) be the Boolean algebra of all idempotents in L 0 (Ω). By ∇ we denote an arbitrary complete Boolean subalgebra of ∇(Ω).
Let E be a linear space over the real field R. By ∥ · ∥ we denote a L 0 (Ω)-valued norm on E. Then the pair (E, ∥ · ∥) is called a lattice-normed space (LNS) over L 0 (Ω). An LNS E is said to be d-decomposable if for every x ∈ E and the decomposition ∥x∥ = f + g with f and g disjoint positive elements in L 0 (Ω) there exist y, z ∈ E such that x = y + z with ∥y∥ = f , ∥z∥ = g.

An even continuous convex function
One can see that L M (∇, µ) ⊂ L 1 (∇, µ). Define the L 0 -valued Orlicz norm on L M (∇, µ) as follows is a Banach-Kantorovich lattice which is called the Orlicz-Kantorovich lattice associated with the L 0 -valued measure [23].
As in the classical setting, the Orlicz spaces, along with the Orlicz norm ∥ · ∥ M on L M (∇, µ), one can consider the L 0 -valued Luxemburg norm Moreover, the pair (L M (∇, µ), ∥ · ∥ (M ) ) is also a Banach -Kantorovich lattice [24]. Now we mention necessary facts from the theory of measurable bundles of Boolean algebras and Banach spaces (see [18] for more details).
Let (Ω, Σ, λ) be the same as above and X be a mapping assigning an L p -space constructed by a real-valued measure µ ω , i.e. L p (∇ ω , µ ω ) to each point ω ∈ Ω and let be a set of sections. In [12] it has been established that the pair (X, L) is a measurable bundle of Banach lattices and L 0 (Ω, X) is modulo ordered isomorphic to L p (∇, µ).

The "zero-two" law for positive contractions in the Orlicz-Kantorovich spaces
In this section we provide the uniform "zero-two" law for the positive contractions of the Orlicz -Kantorovich lattices L M (∇, µ). Using methods of measurable bundles of Orlicz -Kantorovich lattices.   Using the methods of [13] one can establish the following facts.
where | · | is module of an operator.
By means of the argument of [17,Theorem 4.3] we can prove the following result.