Seppo I Hiltunen
This note aims to clarify the interrelations of certain inequivalently defined Banach spaces denoted by C^i(\barΩ) for a natural number i and a bounded open set Ω. We give some sufficient conditions for the equality of these spaces, and present examples to show that the spaces indeed can be unequal for Ωhaving irregular boundary.
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 prove an inequality of the Loéve-Young type for the Riemann-Stieltjes integrals driven by irregular signals attaining their values in Banach spaces and, as a result, we derive a new theorem on the existence of the Riemann-Stieltjes integrals driven by such signals. Also, for any $p\ge1$ we introduce the space of regulated signals $f:[a,b] \rightarrow W$ ($a<b$ are real numbers and $W$ is a Banach space), which may be uniformly approximated with accuracy $δ>0$ by signals whose total variation is of order $δ^{1-p}$ as $δ\rightarrow 0+$ and prove that they satisfy the assumptions of the theore…
If we change the upper and lower density in the definition of distributional chaos of a continuous linear operator on Banach space by the Banach upper and Banach lower density, respectively, we obtain Li-Yorke chaos. Motivated by this fact, we introduce the notions of reiterative distributional chaos of types $1$, $1^+$ and $2$ for continuous linear operators on Banach spaces, which are characterized in terms of the existence of an irregular vector with additional properties. Moreover, we study its relations with other dynamical properties and give conditions for the existence of a vector subs…
Density properties of orbits for a hypercyclic operator on a Banach space
研究Banach空间上超循环算子轨道的密度性质，证明四种情况中仅有一种成立，并通过例子展示所有情况均可能出现。
Mean Li-Yorke chaos in Banach spaces
本文研究Banach空间上算子的平均Li-Yorke混沌，证明其与第二类分布混沌不同，给出等价刻画与判据，并推广到C_0-半群。
On strong Arens irregularity of projective tensor product of Hilbert-Schmidt space
本文研究了Hilbert-Schmidt空间的投影张量积的强Arens不规则性，通过超滤子极限的概念证明其不规则性并非强，并给出了相关函数的性质及A**代数的消去性质。
In this note we provide examples that show that a common notion of causality for linear operators on Banach spaces does not carry over to the closure of the respective operators. We provide an alternative definition for causality, which is equivalent to the usual definition for closed linear operators but does carry over to the closure.
Let C be a nonempty closed convex subset of a Banach space $E$. In this paper we introduce a new definition for variational inequality V I (C, B) on E that generalizes the analogue definition on Hilbert spaces. We generalize (u, v)-cocoercive mappings and v-strongly monotone mappings from Hilbert spaces to Banach spaces. Then we prove the generalized variational inequality V I (C, B) is singleton for (u, v)-cocoercive mappings under appropriate assumptions on Banach spaces that extends and improves [S. Saeidi, Comments on relaxed (u, v)-cocoercive mappings. Int. J. Nonlinear Anal. Appl. 1 (201…
Classification of ill-posedness for bounded linear operators in Banach spaces
针对Banach空间中线性算子不适定性定义不统一的问题，梳理现有相关定义，完成I型、II型不适定性分类并讨论边界混合情形与算子实例。
Motivated by the equivalent definition of a continuous operator between Banach spaces in terms of weakly null nets, we introduce unbounded continuous operators by replacing weak convergence with the unbounded absolutely weak convergence ( $uaw$-convergence) in the definition of a continuous operator between Banach lattices. We characterize order continuous Banach lattices and reflexive Banach lattices in terms of these spaces of operators. Moreover, motivated by characterizing of a reflexive Banach lattice in terms of unbounded absolutely weakly Cauchy sequences, we consider pre-unbounded oper…
We introduce the class of Orlicz-Pettis polynomials between Banach spaces, defined by their action on weakly unconditionally Cauchy series. We give a number of equivalent definitions, examples and counterexamples which highlight the differences between these polynomials and the corresponding linear operators.
The question is addressed of when a Sobolev type space, built upon a general rearrangement-invariant norm, on an $n$-dimensional domain, is a Banach algebra under pointwise multiplication of functions. A sharp balance condition among the order of the Sobolev space, the strength of the norm, and the (ir)regularity of the domain is provided for the relevant Sobolev space to be a Banach algebra. The regularity of the domain is described in terms of its isoperimetric function. Related results on the boundedness of the multiplication operator into lower-order Sobolev type spaces are also establishe…
Banach function spaces done right
本文讨论了（拟）Banach 函数空间的定义，推崇 Zaanen 和 Luxemburg 避免后续定义缺陷的原始定义，并在拟 Banach 函数空间设定下证明了著名基本性质的版本。
A finite-dimensional analogue of the known Gordon-Lewis constant of a Banach space X is introduced; in its definition are used only finite rank operators. It is shown that there exist Banach spaces such that the standard Gordon-Lewis constant of X is finite and its finite-dimensional analogue GLfin(X) is infinite. Moreover, for any Banach space X of cotype 2 its finite-dimensional constant GLfin(X) is finite too.
Compactness in Combinatorial Banach Spaces and Beyond
本文研究组合Banach空间及其扩展中的紧性，提出紧性判据，并揭示其与超图理论的关联。
Let $A$ be a Banach algebra and $A^{**}$ be the second dual of it. We define $\tilde{Z}_1(A^{**})$ as a weak topological center of $A^{**}$ with respect to first Arens product and find some relations between it and the topological center of $A^{**}$. We also extend this new definition into module actions and find relationship between weak topological center of module actions and reflexivity or Arens regularity of some Banach algebras, and we investigate some applications of this new definition in the weak amenability of some Banach algebras.
Let $A$ be a Banach algebra and $A^{**}$ be the second dual of it. We define $\tilde{Z}_1(A^{**})$ as a weak topological center of $A^{**}$ with respect to first Arens product and find some relations between it and the topological center of $A^{**}$. We also extend this new definition into the module actions and find relationship between weak topological center of module actions and reflexivity or Arens regularity of some Banach algebras, and we investigate some applications of this new definition in the weak amenability of some Banach algebras.
A note on the stability of Arens products under small perturbations of multiplication
本文研究Banach代数乘法小扰动对双对偶上Arens积的影响，证明Arens不规则性在足够小扰动下稳定，且等价于一致代数的Banach函数代数的Arens正则性在小扰动下保持稳定。
Let $A$ be a Banach algebra and $A^{**}$ be the second dual of it. We define $\tilde{Z}_1(A^{**})$ as a weak topological center of $A^{**}$ with respect to first Arens product and we will find some relations between this concept and the topological center of $A^{**}$. We also extend this new definition into the module actions and find relationship between weak topological center of module actions and reflexivity or Arens regularity of some Banach algebras, and we investigate some applications of this new definition in the weak amenability of some Banach algebras.
Let C be a nonempty closed convex subset of a real normed linear space $E$ and u, v are positive numbers. In this paper we introduce some new definitions that generalize the analogue definitions from real Hilbert spaces to real normed linear spaces. Indeed, we generalize (u, v)-cocoercive mappings and v-strongly monotone mappings and V I (C, B) for a mapping $B$, from real Hilbert spaces to real normed linear spaces. Then we prove that the generalized variational inequality V I (C, B) is singleton for (u, v)-cocoercive mappings under appropriate assumptions on Banach spaces that extends and im…
It is known that for Banach valued functions there are several approaches to define a Sobolev class. We compare the usual definition via weak derivatives with the Reshetnyak-Sobolev space and with the Newtonian space; in particular, we provide sufficient conditions when all three agree. As well we revise the difference quotient criterion and the property of Lipschitz mapping to preserve Sobolev space when it acting as a superposition operator.
We give a simple definition of a spectral shift function for pairs of nonpositive operators on Banach spaces and prove trace formulas of Lifshitz-Kre\uın type for a perturbation of an operator monotonic (negative complete Bernstein) function of negative and nonpositive operators on Banach spaces induced by nuclear perturbation of an operator argument. The Lipschitzness of such functions is also investigated. The results may be regarded as a contribution to a perturbation theory for Hirsch functional calculus.
On $α$-monotone operators and their resolvent in Banach spaces
本文在实2-一致凸且光滑的Banach空间中引入$\alpha$-单调算子的新定义，建立其结构性质并推广预解式的非扩张性，进而证明前向-反射-后向分裂算法的强收敛性与$R$-线性收敛速率。
Operator ordering by ill-posedness in Hilbert and Banach spaces
本文提出一种基于有界算子级联的算子不适定性排序方法，证明其与 Mathé 和 Hofmann 定义的等价性，并将其推广至 Banach 空间及非线性问题。
We consider real spaces only. Definition. An operator $T:X\to Y$ between Banach spaces $X$ and $Y$ is called a Hahn-Banach operator if for every isometric embedding of the space $X$ into a Banach space $Z$ there exists a norm-preserving extension $\tilde T$ of $T$ to $Z$. A geometric property of Hahn-Banach operators of finite rank acting between finite-dimensional normed spaces is found. This property is used to characterize pairs of finite-dimensional normed spaces $(X,Y)$ such that there exists a Hahn-Banach operator $T:X\to Y$ of rank $k$. The latter result is a generalization of a recent …
We study Li-Yorke chaos and distributional chaos for operators on Banach spaces. More precisely, we characterize Li-Yorke chaos in terms of the existence of irregular vectors. Sufficient "computable" criteria for distributional and Li-Yorke chaos are given, together with the existence of dense scrambled sets under some additional conditions. We also obtain certain spectral properties. Finally, we show that every infinite dimensional separable Banach space admits a distributionally chaotic operator which is also hypercyclic.
Tauberian pairs of closed subspaces of a Banach space
本文引入Banach空间闭子空间的Tauberian、cotauberian和弱紧对概念，探讨其理论并应用于Banach空间不可分解性及扩展有界算子定义。
It is an English translation of the paper originally published in Russian and Ukrainian in 1987. In the appendix of his book S.Banach introduced the following definition Let $X$ be a Banach space and $Γ$ be a subspace of the dual space $X^*$. The set of all limits of $w^{*}$-convergent sequences in $Γ$ is called the $w^*${\it -derived set} of $Γ$ and is denoted by $Γ_{(1)}$. For an ordinal $α$ the $w^{*}$-{\it derived set of order} $α$ is defined inductively by the equality: $$ Γ_{(α)}=\bigcup _{β<α}((Γ_{(β)})_{(1)}. $$
Recently, there has been emerging interest in constructing reproducing kernel Banach spaces (RKBS) for applied and theoretical purposes such as machine learning, sampling reconstruction, sparse approximation and functional analysis. Existing constructions include the reflexive RKBS via a bilinear form, the semi-inner-product RKBS, the RKBS with $\ell^1$ norm, the $p$-norm RKBS via generalized Mercer kernels, etc. The definitions of RKBS and the associated reproducing kernel in those references are dependent on the construction. Moreover, relations among those constructions are unclear. We expl…
In this paper we present a rare combination of abstract results on the spectral properties of slanted matrices and some of their very specific applications to frame theory and sampling problems. We show that for a large class of slanted matrices boundedness below of the corresponding operator in $\ell^p$ for some $p$ implies boundedness below in $\ell^p$ for all $p$. We use the established resultto enrich our understanding of Banach frames and obtain new results for irregular sampling problems. We also present a version of a non-commutative Wiener's lemma for slanted matrices.
We give an abstract definition, similar to the axioms of a Stein manifold, of a class of complex Banach manifolds in such a way that a manifold belongs to the class if and only if it is biholomorphic to a closed split complex Banach submanifold of a separable Banach space.
We introduce the numerical spectrum $σ_n(A)\subset \mathbb{C}$ of an (unbounded) linear operator $A$ on a Banach space $X$ and study its properties. Our definition is closely related to the numerical range $W(A)$ of $A$ and always yields a superset of $W(A)$. In the case of bounded operators on Hilbert spaces, the two notions coincide. However, unlike the numerical range, $σ_n(A)$ is always closed, convex and contains the spectrum of $A$. In the paper we strongly emphasise the connection of our approach to the theory of $C_0$-semigroups.
We present a new approach to define a suitable integral for functions with values in quasi-Banach spaces. The integrals of Bochner and Riemann have deficiencies in the non-locally convex setting. The study of an integral for $p$-Banach spaces initiated by Vogt is neither totally satisfactory, since there are quasi-Banach spaces which are $p$-convex for all $0<p<1$, so it is not always possible to choose an optimal $p$ to develop the integration. Our method puts the emphasis on the galb of the space, which permits a precise definition of its convexity. The integration works for all spaces of ga…
The Peetre "plus-minus" interpolation spaces $\left\langle A_{0},A_{1}\right\rangle _θ$ are defined variously via conditions about the unconditional convergence of certain Banach space valued series whose terms have coefficients which are powers of 2 or, alternatively, powers of $e$. It may seem intuitively obvious that using powers of 2, or of $e$, or powers of some other constant number greater than 1 in such definitions should produce the same space to within equivalence of norms. To allay any doubts, we here offer an explicit proof of this fact, via a "continuous" definition of the same sp…
Trichotomy for the orbits of a hypercyclic operator on a Banach space
本文在可分 Banach 空间上建立了超循环算子轨道的三分法，区分均值渐近于零、绝对均值不规则和均值发散到无穷三种情形，并用加权后移算子说明三种情况均可实现。
The concept of Rademacher type $p$ ($1\leq p\leq2$) plays an important role in the local theory of Banach spaces. In \cite{mas88} Mascioni considers a weakening of this concept and shows that for a Banach space $X$ weak Rademacher type $p$ implies Rademacher type $r$ for all $r<p$. As with Rademacher type $p$ and weak Rademacher type $p$, we introduce the concept of Haar type $p$ and weak Haar type $p$ by replacing the Rademacher functions by the Haar functions in the respective definitions. We show that weak Haar type $p$ implies Haar type $r$ for all $r<p$. This solves a problem left open by…
We extend the Gabor analysis in \cite{GaSa} to a broad class of modulation spaces, allowing more general mixed quasi-norm estimates and weights in the definition of the modulation space quasi-norm. For such spaces we deduce invariance and embedding properties, and that the elements admit reconstructible sequence space representations using Gabor frames.
Having as start point the classic definitions of resolvent set and spectrum of a linear bounded operator on a Banach space, we introduce the resolvent set and spectrum of a family of linear bounded operators on a Banach space. In addition, we present some results which adapt to asymptotic case the classic results.
Building ideals of two-Lipschitz operators between mertic and Banach spaces
该研究针对度量空间与Banach空间之间的双Lipschitz算子理想问题，刻画其单射包与对偶理想定义，提出Lipschitz化和因子分解两种构造方法，证明新理想的闭性、单射性与对称性依赖原理想的对应性质并给出示例。
In this paper, we investigate a new notion of accretive mappings called generalized $αβ$-$H((.,.),(.,.))$-mixed accretive mappings in Banach spaces. We extend the concept of proximal-point mappings associated with generalized $m$-accretive mappings to the generalized $αβ$-$H((.,.),(.,.))$-mixed accretive mappings and prove that the proximal-point mapping associated with generalized $αβ$-$H((.,.),(.,.))$-mixed accretive mapping is single-valued and Lipschitz continuous. Some examples are given to justify the definition of generalized $αβ$-$H((.,.),(.,.))$-mixed accretive mappings. Further, by u…
Motivated by the equivalent definition of a continuous operator between Banach spaces in terms of weakly null nets, we introduce two types of continuous operators between Banach lattices using unbounded absolute weak convergence. We characterize reflexive Banach lattices in terms of these spaces of operators. Furthermore, we investigate whether or not the adjoint of these classes of operators has the corresponding property. In addition, we show that these kinds of operators are norm closed but not order closed. Moreover, we establish how these classes of continuous operators are connected to t…
Starting from the classic definitions of local resolvent set and spectrum of a linear bounded operator on a Banach space, we introduce the local resolvent set and spectrum, the local space and the single-valued extention property of a family of linear bounded operators on a Banach space. Keeping the analogy with the classic case, we extend some the known results from the case of a linear bouded operator to the case of a family of linear bounded operators on a Banach space.
The Banach algebras $AC(σ)$ and $BV(σ)$
本文采用简化定义，自足地阐述了Banach代数 $AC(σ)$ 与 $BV(σ)$ 的主要性质，以统一处理良有界算子与三角良有界算子。
We study extension theorems for Lipschitz-type operators acting on metric spaces and with values on spaces of integrable functions. Pointwise domination is not a natural feature of such spaces, and so almost everywhere inequalities and other measure-theoretic notions are introduced.%more appropriate. Thus, we adapt the classical definition of Lipschitz map to the context of spaces of integrable functions by introducing such elements. We analyze Lipschitz type inequalities in two fundamental cases. The first concerns a.e. pointwise inequalities, while the second considers dominations involving …
We define an {\it $(X_1,Θ, X_2)$-frame} with Banach spaces $X_2\subseteq X_1$, $|\cdot|_1 \leq |\cdot|_2$, and a $BK$-space $(Θ, \snorm[\cdot])$. Then by the use of decreasing sequences of Banach spaces ${X_s}_{s=0}^\infty$ and of sequence spaces ${Θ_s}_{s=0}^\infty$, we define a general Fr\' echet frame on the Fr\' echet space $X_F=\bigcap_{s=0}^\infty X_s$. We give frame expansions of elements of $X_F$ and its dual $X_F^*$, as well of some of the generating spaces of $X_F$ with convergence in appropriate norms. Moreover, we give necessary and sufficient conditions for a general pre-Fr\' eche…
We study $C$-rich spaces, lush spaces, and $C$-extremely regular spaces concerning with the Mazur-Ulam property. We show that a uniform algebra and the real part of a uniform algebra with the supremum norm are $C$-rich spaces, hence lush spaces. We prove that a uniformly closed subalgebra of the algebra of complex-valued continuous functions on a locally compact Hausdorff space which vanish at infinity is $C$-extremely regular provided that it separates the points of the underlying space and has no common zeros. In section 3 we exhibit descriptions on the Choquet bounday, the \vSilov bounday, …
For Banach left and right module actions, we extend some propositions from Lau and $\ddot{U}lger$ into general situations and we establish the relationships between topological centers of module actions. We also introduce the new concepts as $Lw^*w$-property and $Rw^*w$-property for Banach $A-bimodule$ $B$ and we obtain some conclusions in the topological center of module actions and Arens regularity of Banach algebras. we also study some factorization properties of left module actions and we find some relations of them and topological centers of module actions. For Banach algebra $A$, we exte…
Isometric, symmetric and isosymmetric commuting $d$-tuples of Banach space operators
该研究将Banach空间算子相关性质推广到交换$d$元组，阐明其诸多结构性质可通过单算子情形的常用论证推导，厘清该推广的内在逻辑。
We introduce a weakened notion of norm attainment for bounded linear operators between Banach spaces which we call \emph{quasi norm attaining operators}. An operator $T\colon X \longrightarrow Y$ between the Banach spaces $X$ and $Y$ is quasi norm attaining if there is a sequence $(x_n)$ of norm one elements in $X$ such that $(Tx_n)$ converges to some $u\in Y$ with $\|u\|=\|T\|$. Norm attaining operators in the usual sense (i.e.\ operators for which there is a point in the unit ball where the norm of its image equals the norm of the operator) and compact operators satisfy this definition. The …
Let $X$ be a separable nonquasireflexive Banach space. Let $Y$ be a Banach space isomorphic to a subspace of $X^*$. The paper is devoted to the following questions: 1. Under what conditions does there exist an isomorphic embedding $T:Y\to X^*$ such that subspace $T(Y)\subset X^*$ is total? 2. If such embeddings exist, what are the possible orders of $T(Y)$? Here we need to recall some definitions. For a subset $M\subset X^*$ we denote the set of all limits of weak$^*$ convergent sequences in $M$ by $M_{(1)}$. Inductively, for ordinal number $α$ we let $$M_{(α)}=\cup_{β<α}(M_{(β)})_{(1)}.$$ The…
Localization of operator-valued frames
本文引入算子值框架的局部化概念，证明其规范对偶保持内在局部化，证明完美重构级数在Hilbert空间及一类（拟）Banach空间中收敛，并应用于不规则Gabor g-框架。
Generalized Reproducing Kernel Banach Spaces: A Functional Analytic Framework for Abstract Neural Networks
本文提出广义再生核巴拿赫空间框架，用于构建抽象神经网络，扩展了传统RKBS在处理巴拿赫值余域问题中的应用，推动了神经架构设计与数学建模的结合。
On the Numerical Range of Linear Relations in Banach Spaces
本文提出适配线性关系多值特性的巴拿赫空间数值域新定义，推导相关谱包含结果与预解估计，将其应用于算子束得到更精确的谱信息，还可用于半群的严格指数衰减分析。
We give, for general Banach spaces, a characterization of the sequential lower limit of maximal monotone operators of type (D) and prove its representability. As a consequence, using a recent extension of the Moreau-Yosida regularization for type (D) operators, we extend to general Banach spaces the definitions of the variational sum of monotone operators and the variational composition of monotone operators with continuous linear mappings, and we prove that both operators are representable.
In this paper we present part I of nonlinear operator ideals theory between metric spaces and Banach spaces. Building upon the definition of operator ideal between arbitrary Banach spaces of A. Pietsch we pose three types of nonlinear versions of operator ideals. We introduce several examples of nonlinear ideals and the relationships between them. For every space ideal $\mathsf{A}$ can be generated by a special nonlinear ideal which consists of those Lipschitz operators admitting a factorization through a Banach space $\mathbf{M}\in\mathsf{A}$. We investigate products and quotients of nonlinea…
We apply the metrical approach to Sobolev spaces, which arise in various evolution PDEs. Functions from those spaces are defined on an interval and take values in a family of Banach spaces. In this case we adapt the definition of Newtonian spaces. For a monotone family, we show the existence of weak derivative, obtain an isomorphism to the standard Sobolev space, and provide some scalar characteristics.
We develop a functional framework suitable for the treatment of partial differential equations and variational problems on evolving families of Banach spaces. We propose a definition for the weak time derivative that does not rely on the availability of a Hilbertian structure and explore conditions under which spaces of weakly differentiable functions (with values in an evolving Banach space) relate to classical Sobolev--Bochner spaces. An Aubin--Lions compactness result is proved. We analyse concrete examples of function spaces over time-evolving spatial domains and hypersurfaces for which we…
We extend the classical Brunel-Sucheston definition of the spreading model by introducing the $\mathcal{F}$-sequences $(x_s)_{s\in\mathcal{F}}$ in a Banach space and the plegma families in $\mathcal{F}$ where $\mathcal{F}$ is a regular thin family. The new concept yields a transfinite increasing hierarchy of classes of spreading sequences. We explore the corresponding theory and we present examples establishing this hierarchy and illustrating the limitation of the theory.
Rademacher type and Enflo type coincide
本文证明了Banach空间的Rademacher型与Enflo型重合，解决了该领域长期开放的问题，其证明基于离散立方体上Pisier不等式的新型无维度类似物。
The relation $xy-yx=h(y)$, where $h$ is a holomorphic function, occurs naturally in the definitions of some quantum groups. To attach a rigorous meaning to the right-hand side of this equality, we assume that $x$ and $y$ are elements of a Banach algebra (or of an Arens--Michael algebra). We prove that the universal algebra generated by a commutation relation of this kind can be represented explicitly as an analytic Ore extension. An analysis of the structure of the algebra shows that the set of holomorphic functions of $y$ degenerates, but at each zero of $h$, some local algebra of power serie…
We give a definition of the spectral flow for continuous paths in the space of bounded and essentially hyperbolic operators. We provide a homotopical characterization of the spectral flow in terms of a group homomorphism of the fundamental group of the projectors of the Calkin algebra with the infinite cyclic group Z. This characterization helps us to exhibit examples of infinite-dimensional Banach spaces where the spectral flow is not injective nor surjective. We prove that a path with spectral flow equal to an integer m exists if and only if there exists a projector P connected by an arc to …
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。