### A compactness criterion of mixed Krasnoselskiĭ-Riesz type in regular ideal spaces of vector functions.

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Let $E$ be a holomorphic Banach bundle over a compact complex manifold, which can be defined by a cocycle of holomorphic transition functions with values of the form $\mathrm{id}+K$ where $K$ is compact. Assume that the characteristic fiber of $E$ has the compact approximation property. Let $n$ be the complex dimension of $X$ and $0\le q\le n$. Then: If $V\to X$ is a holomorphic vector bundle (of finite rank) with ${H}^{q}(X,V)=0$, then $dim{H}^{q}(X,V\otimes E)\<\infty $. In particular, if $dim{H}^{q}(X,\mathcal{O})=0$, then $dim{H}^{q}(X,E)\<\infty $.

We give a sufficient and necessary condition for a Radon-Nikodým compact space to be Eberlein compact in terms of a separable fibre connecting weak-* and norm approximation.

We provide a new proof of James' sup theorem for (non necessarily separable) Banach spaces. One of the ingredients is the following generalization of a theorem of Hagler and Johnson: "If a normed space E does not contain any asymptotically isometric copy of l1, then every bounded sequence of E' has a normalized l1-block sequence pointwise converging to 0".

A quantitative version of Krein's Theorem on convex hulls of weak compact sets is proved. Some applications to weakly compactly generated Banach spaces are given.

We give a characterization of $K$-weakly precompact sets in terms of uniform Gateaux differentiability of certain continuous convex functions.

Given an operator ideal , we say that a Banach space X has the approximation property with respect to if T belongs to ${\overline{S\circ T:S\in \mathcal{F}\left(X\right)}}^{{\tau}_{c}}$ for every Banach space Y and every T ∈ (Y,X), ${\tau}_{c}$ being the topology of uniform convergence on compact sets. We present several characterizations of this type of approximation property. It is shown that some of the existing approximation properties in the literature may be included in this setting.

∗ Supported by Research grants GAUK 190/96 and GAUK 1/1998We prove that the dual unit ball of the space C0 [0, ω1 ) endowed with the weak* topology is not a Valdivia compact. This answers a question posed to the author by V. Zizler and has several consequences. Namely, it yields an example of an affine continuous image of a convex Valdivia compact (in the weak* topology of a dual Banach space) which is not Valdivia, and shows that the property of the dual unit ball being Valdivia is not an isomorphic...

The paper contains some applications of the notion of $\u0141$ sets to several classes of operators on Banach lattices. In particular, we introduce and study the class of order $\left(\mathrm{L}\right)$-Dunford-Pettis operators, that is, operators from a Banach space into a Banach lattice whose adjoint maps order bounded subsets to an $\left(\mathrm{L}\right)$ sets. As a sequence characterization of such operators, we see that an operator $T:X\to E$ from a Banach space into a Banach lattice is order $\u0141$-Dunford-Pettis, if and only if $\left|T\right({x}_{n}\left)\right|\to 0$ for $\sigma (E,{E}^{\text{'}})$ for every weakly null...

We consider discrete versions of Morrey spaces introduced by Gunawan et al. in papers published in 2018 and 2019. We prove continuity and compactness of multiplication operators and commutators acting on them.

We give a basic sequence characterization of relative weak compactness in c₀ and we construct new examples of closed, bounded, convex subsets of c₀ failing the fixed point property for nonexpansive self-maps. Combining these results, we derive the following characterization of weak compactness for closed, bounded, convex subsets C of c₀: such a C is weakly compact if and only if all of its closed, convex, nonempty subsets have the fixed point property for nonexpansive mappings.

Let Ω be a bounded domain in Rn and denote by idΩ the restriction operator from the Besov space Bpq1+n/p(Rn) into the generalized Lipschitz space Lip(1,-α)(Ω). We study the sequence of entropy numbers of this operator and prove that, up to logarithmic factors, it behaves asymptotically like ek(idΩ) ~ k-1/p if α > max (1 + 2/p + 1/q, 1/p). Our estimates improve previous results by Edmunds and Haroske.

For p ≥ 1, a subset K of a Banach space X is said to be relatively p-compact if $K\subset {\sum}_{n=1}^{\infty}\alpha \u2099x\u2099:\alpha \u2099\in Ball\left({l}_{{p}^{\text{'}}}\right)$, where p’ = p/(p-1) and $x\u2099\in {l}_{p}^{s}\left(X\right)$. An operator T ∈ B(X,Y) is said to be p-compact if T(Ball(X)) is relatively p-compact in Y. Similarly, weak p-compactness may be defined by considering $x\u2099\in {l}_{p}^{w}\left(X\right)$. It is proved that T is (weakly) p-compact if and only if T* factors through a subspace of ${l}_{p}$ in a particular manner. The normed operator ideals $({K}_{p},{\kappa}_{p})$ of p-compact operators and $({W}_{p},{\omega}_{p})$ of weakly p-compact operators, arising from these factorizations,...

We provide examples of nonseparable compact spaces with the property that any continuous image which is homeomorphic to a finite product of spaces has a maximal prescribed number of nonseparable factors.

The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for $L\xb2\left({\mathbb{R}}^{d}\right)$, and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov-Triebel-Lizorkin, modulation and Bargmann-Fock spaces.

We study compactness and related topological properties in the space L¹(m) of a Banach space valued measure m when the natural topologies associated to convergence of vector valued integrals are considered. The resulting topological spaces are shown to be angelic and the relationship of compactness and equi-integrability is explored. A natural norming subset of the dual unit ball of L¹(m) appears in our discussion and we study when it is a boundary. The (almost) complete continuity of the integration...

In this paper the author proved the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with variable exponent. As an application he proved the boundedness of certain sublinear operators on the weighted variable Lebesgue space. The proof of the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent does not contain any mistakes. But in the proof of the boundedness of certain sublinear operators on the weighted...

We consider the compact spaces σₙ(Γ) of subsets of Γ of cardinality at most n and their countable products. We give a complete classification of their Banach spaces of continuous functions and a partial topological classification.