Let 1< p_1, … , p_n< ∞ , 1≤ q < ∞ be such that ∑ _i=1^n 1/p_i < 1/q and let μ _1, … , μ _n, ν be arbitrary measures. Generalizing known linear and multilinear results, we prove that all positive n-linear operators from ℓ _p_1×⋯×ℓ _p_n to L_q(ν ) and from L_p_1(μ _1) ×⋯× L_p_n(μ _n) to ℓ _q are compact. This result, along with other related ones concerning free Banach lattices, shall emerge as consequences of some facts we prove about M-weakly compact multilinear operators on Banach lattices.

Let E be a Banach space (or a Banach lattice), let tau be a vector topology on E and let x be a sequence (or a positive sequence) in E not converging to zero with respect to tau. We show how to construct infinite dimensional Banach spaces (or Banach lattices) consisting, up to the origin, of sequences in E not converging to zero with respect to tau and containing a subsequence of x. Plenty of applications to Banach space theory and to Banach lattice theory are provided.

We prove several abstract results giving general conditions under which subspaces of linear or multilinear operators on Banach spaces or Banach lattices are closed. Each of these abstract results is followed by concrete applications, concerning classes of linear/multilinear operators already studied as well as new classes.

Given a Furstenberg family Fand a subset Gamma subset of C, we introduce and explore the notions of (F, Gamma)-supercyclic operator and F-hypercyclic scalar set. First, the study of (F, C)-supercyclic operators yields new interesting information about frequently supercyclic, degrees l'-frequently supercyclic, reiteratively supercyclic and supercyclic operators. Then we provide a criterion for identifying (F, Gamma)-supercyclic operators. As applications of this criterion, we show that any unilateral pseudo-shift operator on c(0)(N) or & ell;(p)(N), 1 <= p < infinity, is (F, Gamma)-supercyclic for any hypercyclicity set F and any unbounded subset Gamma of C. Moreover, under the same condition on Gamma, we show that any separable infinite-dimensional Banach space supports an (F, Gamma)supercyclic operator. Finally, our study provides sufficient and necessary conditions for a subset Gamma subset of C to be an F-hypercyclic scalar set. These results give partial answers to a question raised by Charpentier et al. (2016) [19]. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let B be an unilateral weighted backward shift on ℓ_p, 1 ≤ p < ∞, that admits a 𝒰-frequently hypercyclic subspace. We prove that B admits such a subspace free of frequently hypercyclic vectors. The proof technique we develop also allows us to prove that B admits a hypercyclic subspace free of 𝒰-frequently hypercyclic vectors, and to solve a question posed by Bès and Menet in 2015 on the existence of common 𝒰-frequently hypercyclic subspaces.

Let X1,…,Xm be Banach spaces and let E1,…,Em,F be Banach lattices. Our main results read as follows: (i) The linear adjoint A∗ of a continuous multilinear operator A:X1×⋯×Xm→F is M-weakly compact if and only if A is L-weakly compact. (ii) The linear adjoint A∗ of a multilinear operator of order bounded variation A:E1×⋯×Em→F is L-weakly compact if and only if the linearization of A on the positive projective tensor product is M-weakly compact. In our way to prove these results, we develop the basic theory of linear adjoints of multilinear operators between Riesz spaces, we prove that multilinear operators of order bounded variation between Banach lattices are continuous, and we explore different notions of multilinear operators of M-weakly compact-type.

It is well known that weakly p-summable sequences in a Banach space E are associated to bounded operators from ℓp⁎ to E, and unconditionally p-summable sequences in E are associated to compact operators from ℓp⁎ to E. Generalizing these results to a quite wide environment, we characterize the classes of Banach spaces-valued sequences that are associated to (or represented by) some Banach operator ideal. Using these characterizations, we decide, among all sequence classes that usually appear in the literature, which are represented by some Banach operator ideal and which are not. Moreover, to each class that is represented by some Banach operator ideal, we show an ideal that represents it. Illustrative examples and additional applications are provided.

We give a necessary condition and a sufficient condition on the Banach lattices E and F so that an operator from E to F is DW-compact whenever its adjoint is DW-compact. We do the same, with different conditions, for DW-DP operators. Moreover, we characterize the Banach lattices E and F for which the adjoint of every DW-compact operator from E to F is DW-compact.

We prove the following results: (i) Every absolutely weakly compact set in a Banach lattice is absolutely weakly sequentially compact. (ii) The converse of (i) holds if E is separable or B_E^** is absolutely weak^* compact. (iii) Every absolutely weakly compact subset of a Banach lattice is contained in the closed convex hull of an absolutely weakly null sequence if and only if the Banach lattice has the positive Schur property. Examples and applications are provided.

Let X be a (real or complex) infinite dimensional linear space. We establish conditions on a homogeneous polynomial P on X so that, if W is any finite dimensional subspace of X on which P vanishes, then P vanishes on an infinite dimensional subspace of X containing W. In the complex case, this is a step beyond the classical result due to Plichko and Zagorodnyuk. Applications to the real case are also provided.

Let $E_1, \ldots, E_m$ be (non necessarily Archimedean) Riesz spaces, let $F$ be an Archimedean Riesz space and let $A \colon E_1 \times \cdots \times E_m \to F$ be an order bounded disjointness preserving $m$-linear operator. We prove that all Arens extensions of $A$ are disjointness preserving if either $A$ has finite rank or the spaces are Banach lattices and $F^*$ has a Schauder basis consisting of disjointness preserving functionals.

First we characterize the Banach lattices E whose biduals have the positive Schur property by means of second adjoints of operators on E being almost Dunford-Pettis. Next we extend some known results concerning conditions on the Banach lattices E and F under which the adjoint T & lowast; and the second adjoint T & lowast;& lowast; of any positive almost Dunford-Pettis operator T : E -> F are almost Dunford-Pettis. Finally, we prove when T & lowast; and T & lowast;& lowast; are almost Dunford-Pettis for any (non necessarily almost Dunford-Pettis) T that is either bounded, regular, order bounded or weakly compact.

Let E_1, …, E_m be (non necessarily Archimedean) Riesz spaces, let F be an Archimedean Riesz space and let A E_1 ×⋯× E_m → F be a regular disjointness preserving m-linear operator. We prove that all Arens extensions of A are disjointness preserving if either A has finite rank or the spaces are Banach lattices and F^* has a Schauder basis consisting of disjointness preserving functionals.

The purpose of this paper is to extend to spaces of nonlinear operators, and also to more general spaces of linear operators, a recent result on lineability of sets of non-injective linear operators. We prove, for quite general spaces A of (linear or nonlinear) maps from an arbitrary set to a sequence space, that, for every 0 f ∈ A , the subset of A of non-injective maps contains an infinite dimensional subspace of A containing f. We provide applications to spaces of linear operators between quasi-Banach spaces, to spaces of linear operators belonging to an operator ideal, and, in the nonlinear setting, to spaces of homogeneous polynomials and to spaces of vector-valued Lipschitz functions on metric spaces.

First we give conditions on a Banach lattice $E$ so that an operator $T$ from $E$ to any Banach space is disjoint $p$-convergent if and only if $T$ is almost Dunford-Pettis. Then we study when adjoints of positive operators between Banach lattices are disjoint $p$-convergent. For instance, we prove that the following conditions are equivalent for all Banach lattices $E$ and $F$: (i) A positive operator $T \colon E \to F$ is almost weak $p$-convergent if and only if $T^*$ is disjoint $p$-convergent; (ii) $E^*$ has order continuous norm or $F^*$ has the positive Schur property of order $p$. Very recent results are improved, examples are given and applications of the main results are provided.

We construct a general framework that generates classes of multilinear operators between Banach spaces which encompasses, as particular cases, the several classes of summing type multilinear operators that have been studied individually in the literature. Summing operators by blocks in the isotropic and anisotropic cases are taken into account. The classes we create are shown to be Banach ideals of multilinear operators and applications to coherence and coincidence theorems are provided.