## Displaying similar documents to “A characterization of almost continuity and weak continuity”

### On the class of positive almost weak${}^{☆}$ Dunford-Pettis operators

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we introduce and study the class of almost weak${}^{☆}$ Dunford-Pettis operators. As consequences, we derive the following interesting results: the domination property of this class of operators and characterizations of the wDP${}^{☆}$ property. Next, we characterize pairs of Banach lattices for which each positive almost weak${}^{☆}$ Dunford-Pettis operator is almost Dunford-Pettis.

### Induced almost continuous functions on hyperspaces

Colloquium Mathematicae

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For a metric continuum X, let C(X) (resp., ${2}^{X}$) be the hyperspace of subcontinua (resp., nonempty closed subsets) of X. Let f: X → Y be an almost continuous function. Let C(f): C(X) → C(Y) and ${2}^{f}:{2}^{X}\to {2}^{Y}$ be the induced functions given by $C\left(f\right)\left(A\right)=c{l}_{Y}\left(f\left(A\right)\right)$ and ${2}^{f}\left(A\right)=c{l}_{Y}\left(f\left(A\right)\right)$. In this paper, we prove that: • If ${2}^{f}$ is almost continuous, then f is continuous. • If C(f) is almost continuous and X is locally connected, then f is continuous. • If X is not locally connected, then there exists an almost continuous function f: X → [0,1]...

### The almost Daugavet property and translation-invariant subspaces

Colloquium Mathematicae

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Let G be a metrizable, compact abelian group and let Λ be a subset of its dual group Ĝ. We show that ${C}_{\Lambda }\left(G\right)$ has the almost Daugavet property if and only if Λ is an infinite set, and that $L{¹}_{\Lambda }\left(G\right)$ has the almost Daugavet property if and only if Λ is not a Λ(1) set.

### Almost Prüfer v-multiplication domains and the ring $D+X{D}_{S}\left[X\right]$

Colloquium Mathematicae

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This paper is a continuation of the investigation of almost Prüfer v-multiplication domains (APVMDs) begun by Li [Algebra Colloq., to appear]. We show that an integral domain D is an APVMD if and only if D is a locally APVMD and D is well behaved. We also prove that D is an APVMD if and only if the integral closure D̅ of D is a PVMD, D ⊆ D̅ is a root extension and D is t-linked under D̅. We introduce the notion of an almost t-splitting set. ${D}^{\left(S\right)}$ denotes the ring $D+X{D}_{S}\left[X\right]$, where S is a multiplicatively...

### On some representations of almost everywhere continuous functions on ${ℝ}^{m}$

Colloquium Mathematicae

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It is proved that the following conditions are equivalent: (a) f is an almost everywhere continuous function on ${ℝ}^{m}$; (b) f = g + h, where g,h are strongly quasicontinuous on ${ℝ}^{m}$; (c) f = c + gh, where c ∈ ℝ and g,h are strongly quasicontinuous on ${ℝ}^{m}$.

### Characterising weakly almost periodic functionals on the measure algebra

Studia Mathematica

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Let G be a locally compact group, and consider the weakly almost periodic functionals on M(G), the measure algebra of G, denoted by WAP(M(G)). This is a C*-subalgebra of the commutative C*-algebra M(G)*, and so has character space, say ${K}_{WAP}$. In this paper, we investigate properties of ${K}_{WAP}$. We present a short proof that ${K}_{WAP}$ can naturally be turned into a semigroup whose product is separately continuous; at the Banach algebra level, this product is simply the natural one induced by the Arens...

### On weakly $\theta$-continuous functions

Matematički Vesnik

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### Functionals on Banach Algebras with Scattered Spectra

Bulletin of the Polish Academy of Sciences. Mathematics

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Let A be a complex, commutative Banach algebra and let ${M}_{A}$ be the structure space of A. Assume that there exists a continuous homomorphism h:L¹(G) → A with dense range, where L¹(G) is a group algebra of the locally compact abelian group G. The main results of this note can be summarized as follows: (a) If every weakly almost periodic functional on A with compact spectra is almost periodic, then the space ${M}_{A}$ is scattered (i.e., ${M}_{A}$ has no nonempty perfect subset). (b) Weakly almost periodic...

### A note on an approximative scheme of finding almost homoclinic solutions for Newtonian systems

Banach Center Publications

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In this work we will be concerned with the existence of almost homoclinic solutions for a Newtonian system $q̈+{\nabla }_{q}V\left(t,q\right)=f\left(t\right)$, where t ∈ ℝ, q ∈ ℝⁿ. It is assumed that a potential V: ℝ × ℝⁿ → ℝ is C¹-smooth and its gradient map ${\nabla }_{q}V:ℝ×ℝⁿ\to ℝⁿ$ is bounded with respect to t. Moreover, a forcing term f: ℝ → ℝⁿ is continuous, bounded and square integrable. We will show that the approximative scheme due to J. Janczewska (see [J2]) for a time periodic potential extends to our case.

### On the condition of Λ-convexity in some problems of weak continuity and weak lower semicontinuity

Colloquium Mathematicae

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We study the functional ${I}_{f}\left(u\right)={\int }_{\Omega }f\left(u\left(x\right)\right)dx$, where u=(u₁, ..., uₘ) and each ${u}_{j}$ is constant along some subspace ${W}_{j}$ of ℝⁿ. We show that if intersections of the ${W}_{j}$’s satisfy a certain condition then ${I}_{f}$ is weakly lower semicontinuous if and only if f is Λ-convex (see Definition 1.1 and Theorem 1.1). We also give a necessary and sufficient condition on ${{W}_{j}}_{j=1,...,m}$ to have the equivalence: ${I}_{f}$ is weakly continuous if and only if f is Λ-affine.

### Characterizing matrices with $𝐗$-simple image eigenspace in max-min semiring

Kybernetika

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A matrix $A$ is said to have $𝐗$-simple image eigenspace if any eigenvector $x$ belonging to the interval $𝐗=\left\{x:\underline{x}\le x\le \overline{x}\right\}$ is the unique solution of the system $A\otimes y=x$ in $𝐗$. The main result of this paper is a combinatorial characterization of such matrices in the linear algebra over max-min (fuzzy) semiring. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability)...

### Weak precompactness and property (V*) in spaces of compact operators

Colloquium Mathematicae

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We give sufficient conditions for subsets of compact operators to be weakly precompact. Let ${L}_{w*}\left(E*,F\right)$ (resp. ${K}_{w*}\left(E*,F\right)$) denote the set of all w* - w continuous (resp. w* - w continuous compact) operators from E* to F. We prove that if H is a subset of ${K}_{w*}\left(E*,F\right)$ such that H(x*) is relatively weakly compact for each x* ∈ E* and H*(y*) is weakly precompact for each y* ∈ F*, then H is weakly precompact. We also prove the following results: If E has property (wV*) and F has property (V*), then ${K}_{w*}\left(E*,F\right)$ has property (wV*). Suppose...

### The Lindelöf property in Banach spaces

Studia Mathematica

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A topological space (T,τ) is said to be fragmented by a metric d on T if each non-empty subset of T has non-empty relatively open subsets of arbitrarily small d-diameter. The basic theorem of the present paper is the following. Let (M,ϱ) be a metric space with ϱ bounded and let D be an arbitrary index set. Then for a compact subset K of the product space ${M}^{D}$ the following four conditions are equivalent: (i) K is fragmented by ${d}_{D}$, where, for each S ⊂ D, ${d}_{S}\left(x,y\right)=sup\varrho \left(x\left(t\right),y\left(t\right)\right):t\in S$. (ii) For each countable subset...

### Factorizations of normality via generalizations of $\beta$-normality

Mathematica Bohemica

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The notion of $\beta$-normality was introduced and studied by Arhangel’skii, Ludwig in 2001. Recently, almost $\beta$-normal spaces, which is a simultaneous generalization of $\beta$-normal and almost normal spaces, were introduced by Das, Bhat and Tartir. We introduce a new generalization of normality, namely weak $\beta$-normality, in terms of $\theta$-closed sets, which turns out to be a simultaneous generalization of $\beta$-normality and $\theta$-normality. A space $X$ is said to be weakly $\beta$-normal (w$\beta$-normal$\right)$ if for every...

### On Hattori spaces

Commentationes Mathematicae Universitatis Carolinae

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For a subset $A$ of the real line $ℝ$, Hattori space $H\left(A\right)$ is a topological space whose underlying point set is the reals $ℝ$ and whose topology is defined as follows: points from $A$ are given the usual Euclidean neighborhoods while remaining points are given the neighborhoods of the Sorgenfrey line. In this paper, among other things, we give conditions on $A$ which are sufficient and necessary for $H\left(A\right)$ to be respectively almost Čech-complete, Čech-complete, quasicomplete, Čech-analytic and weakly separated...

### A uniqueness result for the continuity equation in two dimensions

Journal of the European Mathematical Society

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We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation ${\partial }_{t}u+\frac{.}{\stackrel{˙}{}}\left(bu\right)=0$ admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential $f$ associated to $b$. As a corollary we obtain uniqueness under the assumption that the curl of $b$ is a measure. This result can be extended to certain non-autonomous vector fields $b$ with...

### Vitali Lemma approach to differentiation on a time scale

Studia Mathematica

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A new approach to differentiation on a time scale is presented. We give a suitable generalization of the Vitali Lemma and apply it to prove that every increasing function f: → ℝ has a right derivative f₊’(x) for ${\mu }_{\Delta }$-almost all x ∈ . Moreover, ${\int }_{\left[a,b\right)}f{₊}^{\text{'}}\left(x\right)d{\mu }_{\Delta }\le f\left(b\right)-f\left(a\right)$.

### On locally $S$-closed spaces

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Si studiano le condizioni sotto cui l’immagine (o l'immagine inversa) di uno spazio localmente $S$-chiuso sia localmente $S$-chiuso.

### Representing non-weakly compact operators

Studia Mathematica

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For each S ∈ L(E) (with E a Banach space) the operator R(S) ∈ L(E**/E) is defined by R(S)(x** + E) = S**x** + E(x** ∈ E**). We study mapping properties of the correspondence S → R(S), which provides a representation R of the weak Calkin algebra L(E)/W(E) (here W(E) denotes the weakly compact operators on E). Our results display strongly varying behaviour of R. For instance, there are no non-zero compact operators in Im(R) in the case of ${L}^{1}$ and C(0,1), but R(L(E)/W(E)) identifies isometrically...