Research Article On the Lattice Properties of Almost L-Weakly and Almost M- Weakly Compact Operators Barış Akay 1 and Ömer Gök 2 1 Department of Mathematics, Science Faculty, Istanbul University, Istanbul 34134, Turkey 2 Department of Mathematics, Faculty of Arts and Sciences, Yildiz Technical University, Istanbul 34220, Turkey Correspondence should be addressed to Barış Akay; baris.akay@istanbul.edu.tr Received 13 April 2021; Revised 3 May 2021; Accepted 7 May 2021; Published 20 May 2021 Academic Editor: Calogero Vetro Copyright © 2021 Barış Akay and Ömer Gök. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We establish the domination property and some lattice approximation properties for almost L-weakly and almost M-weakly compact operators. Then, we consider the linear span of positive almost L-weakly (resp., almost M-weakly) compact operators and give results about when they form a Banach lattice and have an order continuous norm. 1. Introduction and Notation In this article, we denote real Banach spaces by X and Y and real Banach lattices by E and F . The closed unit ball and the norm dual of X are denoted by B X and X ′ , respectively. E + denotes the positive cone of E, i.e., E + = fx ∈ E : x ≥ 0g. Let x ∈ E. The positive part, the negative part, and the modulus of x are given by x + = x∨0, x - = ð-xÞ∨0, and jxj = ð-xÞ∨x, respectively. For all x, y ∈ E with x ≤ y, the order interval between x and y is denoted by ½x, y = fz ∈ E : x ≤ z ≤ yg . We write SolðAÞ for the solid hull of a set A ⊆ E. By an oper- ator T : X ⟶ Y , we mean a bounded linear mapping. The space of all operators from X into Y is denoted by LðX, Y Þ . If T : X ⟶ Y is an operator, its adjoint T ′ : Y ′ ⟶ X ′ is defined by ðT ′ f ÞðxÞ = f ðTxÞ for each f ∈ Y ′ and for each x ∈ X. The space of all regular operators from E into F is denoted by L r ðE, F Þ. If T : E ⟶ F is an operator with mod- ulus, then the regular norm of T is given by kT k r = kjT jk. For any unexplained notion and terminology, we refer to [1, 2]. Recently, the class of compact and related operators was studied extensively (for instance in [3, 4]). In [3], some identities and estimates for the Hausdorff measures of non- compactness of some operators on the fractional sets of sequences of fractional orders were established, and some classes of compact operators on the fractional sets of sequences were characterized. Also, necessary and sufficient conditions for the class of compact matrix operators from the fractional sets of sequences into the set of bounded sequences were given. In [4], power bounded m-isometric Banach space operator was shown to be polaroid, and the polaroid property for n-quasi left m-invertible operators was proved. In approximation theory, many authors studied some estimates on the positive linear operators with an emphasis on the Kantorovich operators, Durrmeyer- Bernstein operators, and exponential type operators [5–7]. These types of operators have nice and interesting conver- gence properties. The approximation process by the sequence of positive linear operators for integrable or contin- uous functions was presented [5–7]. The common properties of these studies and the present work are positivity of linear operators, uniform convergence of sequences, and Banach lattices (e.g., L 1 ½0, 1 and C[0,1]). The class of compact (resp., weakly compact) operators does not satisfy the domination property [1, 2]. In other words, if two positive operators S, T : E ⟶ F between Banach lattices satisfy 0 ≤ S ≤ T and T is compact (resp., weakly compact), then S is not necessarily compact (resp., weakly compact). Also, a compact operator (resp., weakly Hindawi Journal of Function Spaces Volume 2021, Article ID 1755373, 5 pages https://doi.org/10.1155/2021/1755373