Abstract
First the characteristic of monotonicity of any Banach lattice is expressed in terms of the left limit of the modulus of monotonicity of at the point . It is also shown that for Kothe spaces the classical characteristic of monotonicity is the same as the characteristic of monotonicity corresponding to another modulus of monotonicity . The characteristic of monotonicity of Orlicz function spaces and Orlicz sequence spaces equipped with the Luxemburg norm are calculated. In the first case the characteristic is expressed in terms of the generating Orlicz function only, but in the sequence case the formula is not so direct. Three examples show why in the sequence case so direct formula is rather impossible. Some other auxiliary and complemented results are also presented. By the results of Betiuk-Pilarska and Prus (2008) which establish that Banach lattices with and weak orthogonality property have the weak fixed point property, our results are related to the fixed point theory (Kirk and Sims (2001)).
Highlights
Let us denote S X S X ∩ X, where S X is the unit sphere of a Banach lattice X for its definition, see 1–3 and X is the positive cone of X.A Banach lattice X is said to be strictly monotone X ∈ SM if for all x, y ∈ X such that y ≤ x and y / x we have y < x
It is easy to see that a Banach lattice X is strictly monotone if and only if δm,X 1 1
A Banach lattice X is uniformly monotone if and only if ε0,m X 0. We can define another characteristic of monotonicity of X, namely, ε0,m X sup ε ≥ 0 : ηm,X ε 0 inf ε ≥ 0 : ηm,X ε > 0, 1.4 where ηm,X is the upper modulus of monotonicity defined for all ε > 0 by the formula ηm,X ε inf x y − 1 : x, y ∈ X, x 1, y ≥ ε 1.5 inf x y − 1 : x, y ∈ X, x 1, y ε see 6, 7
Summary
Let us denote S X S X ∩ X , where S X is the unit sphere of a Banach lattice X for its definition, see 1–3 and X is the positive cone of X. We can define another characteristic of monotonicity of X, namely, ε0,m X sup ε ≥ 0 : ηm,X ε 0 inf ε ≥ 0 : ηm,X ε > 0 , 1.4 where ηm,X is the upper modulus of monotonicity defined for all ε > 0 by the formula ηm,X ε inf x y − 1 : x, y ∈ X , x 1, y ≥ ε 1.5 inf x y − 1 : x, y ∈ X , x 1, y ε see 6, 7 It is clear by the triangle inequality for the norm that ηm,X ε ≤ ε for all ε > 0. For more information on the monotonicity properties and coefficient of monotonicity in some Kothe spaces, we refer to 4–14
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