Some properties of almost L-weakly and almost M-weakly compact operators

On the Lattice Properties of Almost L-Weakly and Almost M-Weakly Compact Operators

“…So the norm is order continuous on ½0, I. By Proposition 3.6.19 of [2], we infer that the identity operator I is M-weakly compact, a contradiction ( [12], p. 143).…”

“…Since T is almost M-weakly compact, its adjoint T ′ is almost L-weakly compact. By ( [12], Corollary 4,) T ′ is L-weakly compact, and hence, T is M-weakly compact. Now, suppose (ii) holds.…”

“…Let I : c 0 ⟶ c 0 be the identity operator. Then, I is a positive almost M-weakly compact operator ( [12], p. 143). The order interval ½0, I is the same in AW r M ð E, FÞ and in L r ðE, FÞ (by Theorem 2).…”

“…Note that an operator T : E ⟶ Y is almost Mweakly compact if and only if its adjoint T ′ is almost Lweakly compact, and an operator T : X ⟶ F is almost Lweakly compact whenever its adjoint T ′ is almost Mweakly compact ([11], Theorem 2.5). The relationship between almost L-weakly (resp., almost M-weakly) compact operators and other classes of operators (e.g., compact operators, weakly compact operators, L-weakly, and M-weakly compact operators) was studied in the literature [11][12][13].…”

Some results on almost L-weakly and almost M-weakly compact operators

“…They proved in [4] that an operator T from a Banach space X into a Banach lattice F is almost L-weakly compact if and only if f n (T (x n )) → 0 for every weakly convergent sequence (x n ) of X and every disjoint sequence (f n ) of B F ′ ([4, Theorem 2.2]). After that, A. Elbour et al [6] gave a useful characterization of almost L-weakly compact operator. An operator T from a Banach space X into a Banach lattice F is almost L-weakly compact if and only if T (X) ⊂ F a and f n (T (x n )) → 0 for every weakly null sequence (x n ) of X and every disjoint sequence…”