### A Hölder-type inequality for positive functionals on $\text{\Phi}$-algebras.

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We give a brief survey of recent results of order limited operators related to some properties on Banach lattices.

We characterize Banach lattices on which each regular order weakly compact (resp. b-weakly compact, almost Dunford-Pettis, Dunford-Pettis) operator is AM-compact.

This paper presents an elementary proof and a generalization of a theorem due to Abramovich and Lipecki, concerning the nonexistence of closed linear sublattices of finite codimension in nonatomic locally solid linear lattices with the Lebesgue property.

The paper contains some applications of the notion of $\u0141$ sets to several classes of operators on Banach lattices. In particular, we introduce and study the class of order $\left(\mathrm{L}\right)$-Dunford-Pettis operators, that is, operators from a Banach space into a Banach lattice whose adjoint maps order bounded subsets to an $\left(\mathrm{L}\right)$ sets. As a sequence characterization of such operators, we see that an operator $T:X\to E$ from a Banach space into a Banach lattice is order $\u0141$-Dunford-Pettis, if and only if $\left|T\right({x}_{n}\left)\right|\to 0$ for $\sigma (E,{E}^{\text{'}})$ for every weakly null...

Asymptotic convergence theorems for semigroups of nonnegative operators on a Banach lattice, on C(X) and on ${L}^{p}\left(X\right)$ (1 ≤ p ≤ ∞) are proved. The general results are applied to a class of semigroups generated by some differential equations.

Asymptotic convergence theorems for nonnegative operators on Banach lattices, on ${L}^{\infty}$, on C(X) and on ${L}^{p}(1\le p<\infty )$ are proved. The general results are applied to a class of integral operators on L¹.

A new criterion of asymptotic periodicity of Markov operators on ${L}^{1}$, established in [3], is extended to the class of Markov operators on signed measures.

New sufficient conditions for asymptotic stability of Markov operators are given. These criteria are applied to a class of Volterra type integral operators with advanced argument.