On the class of almost L-weakly and almost M-weakly compact operators

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

“…Assuming that F is Dedekind complete, we have the following result for almost M-weakly compact operators. The proof is similar to the proof of the above theorem with the corresponding results for almost M-weakly compact operators (Theorem 2 and Proposition 2.1 (2) of [11]). + and let ð f α Þ be a net in E ′ such that f α ↓0.…”

“…The class of almost L-weakly (resp., almost M-weakly) compact operators was introduced in [11] as a generalization of that of L-weakly (resp., M-weakly) compact operators. Recall that an operator T : X ⟶ F is called almost Lweakly compact if T maps relatively weakly compact subsets of X onto L-weakly compact subsets of F, and an operator T : E ⟶ Y is called almost M-weakly compact if for each disjoint sequence ðx n Þ in B E and for each weakly convergent sequence ð f n Þ in Y ′ , we have f n ðTx n Þ ⟶ 0.…”

“…Recall that an operator T : X ⟶ F is called almost Lweakly compact if T maps relatively weakly compact subsets of X onto L-weakly compact subsets of F, and an operator T : E ⟶ Y is called almost M-weakly compact if for each disjoint sequence ðx n Þ in B E and for each weakly convergent sequence ð f n Þ in Y ′ , we have f n ðTx n Þ ⟶ 0. Every L-weakly (resp., M-weakly) compact operator is almost L-weakly (resp., almost M-weakly) compact, but the converse is not true in general [11]. For example, the identity operator I : ℓ 1 ⟶ ℓ 1 is almost L-weakly compact but not L-weakly compact, and the identity operator I : ℓ ∞ ⟶ ℓ ∞ is almost M-weakly compact but not M-weakly compact ( [11], p. 1435).…”

“…Every L-weakly (resp., M-weakly) compact operator is almost L-weakly (resp., almost M-weakly) compact, but the converse is not true in general [11]. For example, the identity operator I : ℓ 1 ⟶ ℓ 1 is almost L-weakly compact but not L-weakly compact, and the identity operator I : ℓ ∞ ⟶ ℓ ∞ is almost M-weakly compact but not M-weakly compact ( [11], p. 1435). 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).…”

“…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].…”