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In the first part of this paper, we present some investigations on the class of almost (L) limited operators. We show that an operator T : X → E , from a Banach space X to a Banach lattice E, is almost (L) limited iff its adjoint carries disjoint almost L-sequences to norm null ones. In addition, we improve several results obtained by Oughajji et al. In its second part, we study the relationship between the class of weakly precompact operators and that of order weakly compact (resp. b-weakly compact) operators. Among other things, we show that for a Banach lattice E and a Banach space X the following statements are equivalent: Every order weakly compact (resp. b-weakly compact) operator T : E → X is weakly precompact; The norm of E ′ is order continuous or X does not contain any isomorphic copy of ℓ 1 . [ABSTRACT FROM AUTHOR]

Given a topological dynamical system (X,T), i.e., T is a continuous transformation on a compact metric space X), that admits mistakes and an invariant measure, this paper proves the Brin–Katok formula in this case. In particular, this paper finds that it suffices to study a mistake function which is linear in n and monotonic with respect to Δ. Consequently, this paper shows the Brin–Katok formula for the mean Bowen ball replacing the Bowen metrics with the mean Bowen metrics. The results presented in this paper may have some applications in the study of other properties of a dynamical system which admits small errors. [ABSTRACT FROM AUTHOR]

For two Banach spaces X and Y, we denote by K(X,Y) (resp. B(X,Y)) the space of all compact (resp. bounded) linear operators from X to Y. In this paper, we show that for 1\leq p,q<\infty, K(\ell _{p},\ell _{q}) is an 8-Lipschitz retract of B(\ell _{p},\ell _{q}). [ABSTRACT FROM AUTHOR]

In this paper, we investigate conditions on a pair of Banach lattices E and F that tells us when every positive almost L-weakly compact (resp. almost M- weakly compact) operator T : E → F is weakly compact. Also, we present some necessary conditions that tells us when every weakly compact operator T : E → F is almost M-weakly compact (resp. almost L-weakly compact). In particular, we will prove that if every weakly compact operator from a Banach lattice E into a Banach space X is almost L-weakly compact, then E is a KB-space or X has the Dunford-Pettis property and the norm of E is order continuous. [ABSTRACT FROM AUTHOR]

The primary objective of this paper is to study the problem concerning isomorphic embeddings of ideals of compact operators due to Arazy (see [5, Problem C] and [6, Problem (B)]), which is closely related to Pełczyński's problem of uniqueness of symmetric structure in such ideals. In particular, we show that for a wide class of symmetric sequence spaces E , if C F ↪ S E , then F = ℓ 2 and E has a subspace isomorphic to ℓ 2 , where C X is the symmetric ideal of compact operators on the Hilbert space ℓ 2 induced by a symmetric sequence space X and S X = (⊕ n = 1 ∞ M n) C X . This extends results due to Arazy [5,6]. As a consequence, we contribute to the Arazy–Pełczyński problem on the uniqueness of symmetric structures [5,6]. In particular, we can deal with the case when E has a subspace isomorphic to ℓ 2 or c 0 , which was left untreated in [5,6]. [ABSTRACT FROM AUTHOR]