Stefan Cobzaş
The aim of the present paper is to define compact operators on asymmetric normed spaces and to study some of their properties. The dual of a bounded linear operator is defined and a Schauder type theorem is proved within this framework. The paper contains also a short discussion on various completeness notions for quasi-metric and for quasi-uniform spaces.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We study the geometric structure of compact convex sets in 2-dimensional asymmetric normed lattices. We prove that every q-compact convex set is strongly q-compact and we give a complete geometric description of the compact convex sets with non empty interior in (R^2, q), where q is an asymmetric lattice norm.
The aim of this paper is to present a survey of some recent results obtained in the study of spaces with asymmetric norm. The presentation follows the ideas from the theory of normed spaces (topology, continuous linear operators, continuous linear functionals, duality, geometry of asymmetric normed spaces, compact operators) emphasizing similarities as well as differences with respect to the classical theory. The main difference comes form the fact that the dual of an asymmetric normed space $X$ is not a linear space, but merely a convex cone in the space of all linear functionals on $X.$ Due …
Inspired in the theorem of Krein-Milamn, we investigate the existence of extreme points in compact convex subsets of asymmetric normed spaces. We focus our attention in the finite dimensional case, giving a geometric description of all compact convex subsets of a finite dimensional asymmetric normed space.
We prove that an asymmetric normed space is never a Baire space if the topology induced by the asymmetric norm is not equivalent to the topology of a norm. More precisely, we show that a biBanach asymmetric normed space is a Baire space if and only if it is isomorphic to its associated normed space.
Asymmetric vector norms are generalizations of asymmetric norms, where the subadditivity inequality is understood in ordered vector space sense. This relation imposes strong conditions on the ordering itself. This note studies on these conditions in the general case, and in the case when the asymmetric vector norm is the metric projection onto the cone engendering the order relation.
The paper is concerned with compact bilinear operators on asymmetric normed spaces. The study of multilinear operators on asymmetric normed spaces was initiated by Latreche and Dahia, Colloq. Math. (2020). We go further in this direction and prove a Schauder type theorem on the compactness of the adjoint of a compact bilinear operator and study the ideal properties of spaces of compact bilinear operators. These extend some results of Ramanujan and Schock, Linear and Multilinear Algebra (1985), and Ruch, ibid. (1989), on compact bilinear operators on Banach spaces. On the space of bilinear form…
Let X, Y be asymmetric normed spaces and Lc(X, Y) the convex cone of all linear continuous operators from X to Y. It is known that in general, Lc(X, Y) is not a vector space. The aim of this note is to prove, using the Baire category theorem, that if Lc(X, Y) is a vector space for some asymmetric normed space Y , then X is isomorphic to its associated normed space (the converse is true for every asymmetric normed space Y and is easy to establish). For this, we introduce an index of symmetry of the space X denoted c(X) $\in$ [0, 1] and we give the link between the index c(X) and the fact that L…
There is given the geometric characterization of an asymmetric norm $q$ on the real vector space $X$, for which exists an $u\in X$ such that $q(x-q(x)u)=0$, for each $x\in X$. The result is used in the theory of mutually polar retractions on cones.
Lifted Takahashi Convexity on the Isbell-Convex Hull of an Asymmetrically Normed Real Vector Space
本文在非对称赋范实向量空间的Isbell-凸包上定义了一种提升凸性结构，并证明了其构成一个凸T0-拟度量空间，同时与规范嵌入兼容且与最小函数对的W-凸性相关。
Hahn-Banach type extension results for linear operators on asymmetric normed spaces
本文针对非对称赋范空间上线性算子的延拓问题，将Nachbin的经典思路适配到非对称情形，给出了混合二元交性质与非对称赋范空间单射性等价这一已有结论的直接证明，丰富了非对称赋范空间下的Hahn-Banach型理论。
Bourgain, Brezis & Mironescu showed that (with suitable scaling) the fractional Sobolev $s$-seminorm of a function $f\in W^{1,p}(\mathbb{R}^n)$ converges to the Sobolev seminorm of $f$ as $s\rightarrow1^-$. Ludwig introduced the anisotropic fractional Sobolev $s$-seminorms of $f$ defined by a norm on $\mathbb{R}^n$ with unit ball $K$, and showed that they converge to the anisotropic Sobolev seminorm of $f$ defined by the norm whose unit ball is the polar $L_p$ moment body of $K$, as $s\rightarrow1^-$. The asymmetric anisotropic $s$-seminorms are shown to converge to the anisotropic Sobolev sem…
We present a construction of a compact connected space which supports a normal probability measure.
Under certain initial conditions, we prove the existence of set-valued selectors of univariate compact-valued multifunctions of bounded (Jordan) variation when the notion of variation is defined taking into account only the Pompeiu asymmetric excess between compact sets from the target metric space. For this, we study subtle properties of the directional variations. We show by examples that all assumptions in the main existence result are essential. As an application, we establish the existence of set-valued solutions $X(t)$ of bounded variation to the functional inclusion of the form $X(t)\su…
We construct an LUR-norm on the space C(K) which is lower semicontinuous with respect to the pointwise topology for any Fedorchuk compactum of height 3.
Asymmetric information distances are used to define asymmetric norms and quasimetrics on the statistical manifold and its dual space of random variables. Quasimetric topology, generated by the Kullback-Leibler (KL) divergence, is considered as the main example, and some of its topological properties are investigated.
An inequality of Brascamp and Lieb provides a bound on the covariance of two functions with respect to log-concave measures. The bound estimates the covariance by the product of the $L^2$ norms of the gradients of the functions, where the magnitude of the gradient is computed using an inner product given by the inverse Hessian matrix of the potential of the log-concave measure. Menz and Otto \cite{OM} proved a variant of this with the two $L^2$ norms replaced by $L^1$ and $L^\infty$ norms, but only for $\R^1$. We prove a generalization of both by extending these inequalities to $L^p$ and $L^q$…
A closed densely defined operator $ T $ on a Hilbert space $ \mathcal{H} $ is callled $M$-hyponormal if $\mathcal{D}(T) \subset \mathcal{D}(T^{*}) $ and there exists $ M > 0 $ for which $ \parallel(T-zI)^{*}x \parallel \leq M \parallel(T-zI)x \parallel $ for all $ z \in \mathbb{C}$ and for all $ x\in \mathcal{D}(T)$. In this paper, we prove that if bounded linear operator $ A : \mathcal{H} \rightarrow \mathcal{K}$ is such that $ AB^*\subseteq TA $, where $ B $ is a closed subnormal (resp. a closed $ M $-hyponormal) on $\mathcal{H}$, $ T $ is a closed $ M $-hyponormal (resp. a closed subnormal)…
In this paper, we characterize absolute norm-attainability for compact hyponormal operators. We give necessary and sufficient conditions for a bounded linear compact hyponormal operator on an infinite dimensional complex Hilbert space to be absolutely norm attaining. Moreover, we discuss the structure of compact hyponormal operators when they are self-adjoint and normal. Lastly, we discuss in general, other properties of compact hyponormal operators when they are absolutely norm attaining and their commutators.
We prove a compactness criterion in $L^p(μ,X)$: a subset of $L^p(μ,X)$ is relatively norm compact iff the set of integrals of its functions over any measurable set is relatively norm compact, it satisfies the Fréchet oscillation restriction condition and it is p-uniformly integrable. The proof is elementary.
A linear operator $T$ between two lattice-normed spaces is said to be $p$-compact if, for any $p$-bounded net $x_α$, the net $Tx_α$ has a $p$-convergent subnet. $p$-Compact operators generalize several known classes of operators such as compact, weakly compact, order weakly compact, $AM$-compact operators, etc. Similar to $M$-weakly and $L$-weakly compact operators, we define $p$-$M$-weakly and $p$-$L$-weakly compact operators and study some of their properties. We also study $up$-continuous and $up$-compact operators between lattice-normed vector lattices.
In this paper, we introduce and study a new classes of operators, named AM-unbounded norm compact operators. We study the the basic properties of the new operator and we investigate the lattice-order and topology property of the operator space of AM-unbounded norm compact operator.
In this paper we continue the study of compact-like operators in lattice normed spaces started recently by Aydin, Emelyanov, Erkurşun Özcand and Marabeh. We show among others, that every p-compact operator between lattice normed spaces is p-bounded. The paper contains answers of almost all questions asked by these authors.
In this note we obtain commutativity criteria for normal compact operators using the projective spectrum. We thus improve a corresponding result obtained by Chagouel, Stessin and Zhu[Trans AMS,2016].
General, especially spectral, features of compact normal operators in quaternionic Hilbert spaces are studied and some results are established which generalize well-known properties of compact normal operators in complex Hilbert spaces. More precisely, it is proved that the norm of such an operator always coincides with the maximum of the set of absolute values of the eigenvalues (exploiting the notion of spherical eigenvalue). Moreover the structure of the spectral decomposition of a generic compact normal operator $T$ is discussed also proving a spectral characterization theorem for compact …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。