Vladimir G. Troitsky
We extend the method of minimal vectors to arbitrary Banach spaces. It is proved, by a variant of the method, that certain quasinilpotent operators on arbitrary Banach spaces have hyperinvariant subspaces.
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 analyse several examples of separable Banach spaces, some of them new, and relate them to several dichotomies obtained in the previous paper Banach spaces without minimal subspaces, by classifying them according to which side of the dichotomies they fall. This paper may be seen as a more empirical continuation of Banach spaces without minimal subspaces, in which our stress is on the study of examples for the new classes of Banach spaces considered in that work.
In this paper we study the Pettis integral of fuzzy mappings in arbitrary Banach spaces. We present some properties of the Pettis integral of fuzzy mappings and we give conditions under which a scalarly integrable fuzzy mapping is Pettis integrable.
We study $k-$smoothness of bounded linear operators defined between arbitrary Banach spaces. As an application, we characterize $k-$smooth operators defined from $\ell_1^n$ to an arbitrary Banach space. We also completely characterize $k-$smooth operators defined between arbitrary two-dimensional Banach spaces.
We find a natural class of transformations ("flattened perturbations") of a norming M-basis in a Banach space X, which give a strong norming M-basis in X. This simplifies and generalizes the positive answer to the "strong M-basis problem" solved by P. Terenzi. We also show that in general one cannot achieve uniformly minimality applying standard transformations to a given norming M-basis, despite of the existence in X a uniformly minimal strong M-bases.
Properties of first-order Sobolev-type spaces on abstract metric measure spaces, so-called Newtonian spaces, based on quasi-Banach function lattices are investigated. The set of all weak upper gradients of a Newtonian function is of particular interest. Existence of minimal weak upper gradients in this general setting is proven and corresponding representation formulae are given. Furthermore, the connection between pointwise convergence of a sequence of Newtonian functions and its convergence in norm is studied.
We first include a result of the second author showing that the Banach space S is complementably minimal. We then show that every block sequence of the unit vector basis of S has a subsequence which spans a space isomorphic to its square. By the Pełczyński decomposition method it follows that every basic sequence in S which spans a space complemented in S has a subsequence which spans a space isomorphic to S (i.e. S is a subsequentially prime space).
We establish a relation between Lipschitz operator ideals and linear operator ideals, which fits in the framework of Galois connection between lattices. We use this relationship to give a criterion which allow us to recognize when a Banach Lipschitz operator ideal is of composition type or not. Also, we introduce the concept of minimal Banach Lipschitz operator ideal, which have analogous properties to minimal Banach operator ideals. Also we characterize minimal Banach Lipschitz operator ideals which are of composition type and present examples which are not of this class.
We prove that if $X$ is a quasi-normed space which possesses an infinite countable dimensional subspace with a separating dual, then it admits a strictly weaker Hausdorff vector topology. Such a topology is constructed explicitly. As an immediate consequence, we obtain an improvement of a well-known result of Kalton-Shapiro and Drewnowski by showing that a quasi-Banach space contains a basic sequence if and only if it contains an infinite countable dimensional subspace whose dual is separating. We also use this result to highlight a new feature of the minimal quasi-Banach space constructed by …
Tight-minimal dichotomies in Banach spaces
本文推广Ferenczi-Rosendal方法，证明带容许块系统的Banach空间包含A-紧或A-极小的无穷维子空间，并由此得到基的子序列二分性及紧性的Baire范畴解释。
A Banach space with a Schauder basis is said to be $α$-minimal for some countable ordinal $α$ if, for any two block subspaces, the Bourgain embeddability index of one into the other is at least $α$. We prove a dichotomy that characterises when a Banach space has an $α$-minimal subspace, which contributes to the ongoing project, initiated by W. T. Gowers, of classifying separable Banach spaces by identifying characteristic subspaces.
This paper deals with study of Birkhoff-James orthogonality of a linear operator to a subspace of operators defined between arbitrary Banach spaces. In case the domain space is reflexive and the subspace is finite dimensional we obtain a complete characterization. For arbitrary Banach spaces, we obtain the same under some additional conditions. For arbitrary Hilbert space $ \mathbb{H},$ we also study orthogonality to subspace of the space of linear operators $L(\mathbb{H}), $ both with respect to operator norm as well as numerical radius norm.
Completion problem of upper triangular $3\times3$ operator matrices on arbitrary Banach spaces
本文研究了任意Banach空间上3×3上三角算子矩阵的补全问题，利用Harte的指数定理和Banach空间嵌入技术，推广了Han等人结论，并探讨了高维上三角算子的可逆性条件。
Integral properties of multifunctions with closed convex values are studied. In this more general framework not all the tools and the technique used for weakly compact convex valued multifunctions work. We pay particular attention to the "positive multifunctions". Among them an investigation of multifunctions determined by vector-valued functions is presented. Finally, decomposition results are obtained for scalarly and gauge-defined integrals of multifunctions and a full description of McShane integrability in terms of Henstock and Pettis integrability is given.
The classical Denjoy-Young-Saks theorem on Dini derivatives of arbitrary functions $f: \R \to \R$ was extended by U.S. Haslam-Jones (1932) and A.J. Ward (1935) to arbitrary functions on $\R^2$. This extension gives the strongest relation among upper and lower Hadamard directional derivatives $f^+_H (x,v)$, $f^-_H (x,v)$ ($v \in X$) which holds almost everywhere for an arbitrary function $f:\R^2\to \R$. Our main result extends the theorem of Haslam-Jones and Ward to functions on separable Banach spaces.
If a Banach space is saturated with basic sequences whose linear span embeds into the linear span of any subsequence, then it contains a minimal subspace. It follows that any Banach space is either ergodic or contains a minimal subspace. For a Banach space $X$ with an (unconditional) basis, topological 0-1 law type dichotomies are stated for block-subspaces of $X$ as well as for subspaces of $X$ with a successive FDD on its basis. A uniformity principle for properties of block-sequences, results about block-homogeneity, and a possible method to construct a Banach space with an unconditional ba…
We show that norms on certain Banach spaces $X$ can be approximated uniformly, and with arbitrary precision, on bounded subsets of $X$ by $C^{\infty}$ smooth norms and polyhedral norms. In particular, we show that this holds for any equivalent norm on $c_0(Γ)$, where $Γ$ is an arbitrary set. We also give a necessary condition for the existence of a polyhedral norm on a weakly compactly generated Banach space, which extends a well-known result of Fonf.
We prove the optimal estimate between the Szlenk and $w^*$-dentability indices of an arbitrary $w^*$-compact subset of the dual of a Banach space. For a given $w^*$-compact, convex subset $K$ of the dual of a Banach space, we introduce a two player game the winning strategies of which determine the Szlenk index of $K$. We give applications to the $w^*$-dentability index of a Banach space and of an operator.
A Banach space contains either a minimal subspace or a continuum of incomparable subspaces. General structure results for analytic equivalence relations are applied in the context of Banach spaces to show that if $E_0$ does not reduce to isomorphism of the subspaces of a space, in particular, if the subspaces of the space admit a classification up to isomorphism by real numbers, then any subspace with an unconditional basis is isomorphic to its square and hyperplanes and has an isomorphically homogeneous subsequence.
We prove three new dichotomies for Banach spaces à la W.T. Gowers' dichotomies. The three dichotomies characterise respectively the spaces having no minimal subspaces, having no subsequentially minimal basic sequences, and having no subspaces crudely finitely representable in all of their subspaces. We subsequently use these results to make progress on Gowers' program of classifying Banach spaces by finding characteristic spaces present in every space. Also, the results are used to embed any partial order of size $\aleph_1$ into the subspaces of any space without a minimal subspace ordered by …
Banach spaces with arbitrary finite Baire order
本文构造了可分Banach空间，解决了关于内在Baire阶的开放问题，并推广了Lindenstrauss的结果，表明任何具有基的Banach空间可作为对偶空间中特定Baire类的泛函空间。
We show that for an arbitrary Banach space $E$, the free Banach lattice $FBL[E]$ generated by $E$ satisfies the $σ$-bounded chain condition.
In these notes, we study the relation between uniform and coarse embeddings between Banach spaces. In order to understand this relation better, we also look at the problem of when a coarse embedding can be assumed to be topological. Among other results, we show that if a Banach space $X$ uniformly embeds into a minimal Banach space $Y$, then $X$ simultaneously coarsely and uniformly embeds into $Y$, and if a Banach space $X$ coarsely embeds into a minimal Banach space $Y$, then $X$ simultaneously coarsely and homeomorphically embeds into $Y$ by a map with uniformly continuous inverse.
Relatively recently it was proved that if $Γ$ is an arbitrary set, then any equivalent norm on $c_0(Γ)$ can be approximated uniformly on bounded sets by polyhedral norms and $C^\infty$ smooth norms, with arbitrary precision. We extend this result to more classes of spaces having uncountable symmetric bases, such as preduals of the `discrete' Lorentz spaces $d(w,1,Γ)$, and certain symmetric Nakano spaces and Orlicz spaces. We also show that, given an arbitrary ordinal number $α$, there exists a scattered compact space $K$ having Cantor-Bendixson height at least $α$, such that every equivalent n…
We prove several dichotomies on linear embeddings between Banach spaces. Given an arbitrary Banach space X with a basis, we show that the relations of isomorphism and bi-embedding are meager or co-meager on the Polish set of block-subspaces of X. We relate this result with tightness and minimality of Banach spaces. Examples and open questions are provided.
We study the existence of zeroes of mappings defined in Banach spaces. We obtain, in particular, an extension of the well-known Bolzano-Poincaré-Miranda theorem to infinite dimensional Banach spaces. We also establish a result regarding the existence of periodic solutions to differential equations posed in an arbitrary Banach space.
We prove in this short report that for arbitrary weak converging sequence of sigma-finite Borelian measures in the separable Banach space there is a compact embedded separable subspace such that this measures not only are concentrated in this subspace but weak converge therein.
The main result of this paper states that if a Banach space X has the property that every bounded operator from an arbitrary subspace of X into an arbitrary Banach space of cotype 2 extends to a bounded operator on X, then $B(\ell_{\infty},X^*)=Π_2(\ell_{\infty},X^*)$. If in addition X has the Gaussian average property, then it is of type 2. This implies that the same conclusion holds if X has the Gordon-Lewis property (in particular X could be a Banach lattice) or if X is isomorphic to a subspace of a Banach lattice of finite cotype, thus solving the Maurey extension property for these classe…
On the Banach-Mazur Ellipse
本文通过新方法证明了规范平面中最小巴拿赫-马祖尔距离到单位圆的椭圆特征化，并揭示其与切比雪夫交替的关系。
Extremal Banach-Mazur distance between a symmetric convex body and an arbitrary convex body on the plane
本文证明在平面上，若对称凸体与另一凸体的Banach-Mazur距离达到极值2，则该凸体必为三角形。
This paper considers explicit constructions of Auerbach bases in separable Banach spaces. Answering the question of A. Pełczy{ń}ski, we prove by construction the existence of Auerbach basis in arbitrary subspace of $c_0$ of finite codimension and in the space $C(K)$ for $K$ compact countable metric space.
In this short note, we first consider some inequalities for comparison of some algebraic properties of two continuous algebra-multiplications on an arbitrary Banach space and then, as an application, we consider some very basic observations on the space of all continuous algebra-multiplications on a Banach space.
In this paper, we study the descriptive complexity of some inevitable classes of Banach spaces. Precisely, as shown in [Go], every Banach space either contains a hereditarily indecomposable subspace or an unconditional basis, and, as shown in [FR], every Banach space either contains a minimal subspace or a continuously tight subspace. In these notes, we study the complexity of those inevitable classes as well as the complexity of containing a subspace in any of those classes.
We show that every Banach space saturated with subsymmetric sequences contains a minimal subspace.
We study the class of Banach lattices that are positively polynomially Schur. Plenty of examples and counterexamples are provided, lattice properties of this class are proved, arbitrary $L_p(μ)$-spaces are shown to be positively polynomially Schur, lattice analogues of results on Banach spaces are obtained and relationships with the positive Schur and the weak Dunford-Pettis properties are established.
We introduce a new concept of perturbation of closed linear subspaces and operators in Banach spaces called uniform lambda-adjustment which is weaker than perturbations by small gap, operator norm, q-norm, and K2-approximation. In arbitrary Banach spaces some of the classical Fredholm stability theorems remain true under uniform lambda-adjustment, while other fail. However, uniformly lambda-adjusted subspaces and linear operators retain their (semi--)Fredholm properties in a Banach space which dual is Fréchet-Urysohn in weak* topology. We also introduce another concept of perturbation called u…
Real smooth three-dimensional or higher Banach spaces are isomorphic with respect to the nonlinear structure of Birkhoff-James orthogonality if and only if they are isometrically isomorphic. Moreover, using smooth Radon planes and non-smooth direct sums, in arbitrary dimensions, we construct examples of non-isometric pairs of real Banach spaces that admit norm-preserving homogeneous bicontinuous Birkhoff-James orthogonality preservers among them.
We obtain a pointwise description of functions belonging to function spaces with the lattice property. In particular, it is valid for Banach function spaces provided that the Hardy-Littlewood maximal operator is bounded. We also study weighted grand Sobolev spaces, defined on arbitrary open sets (of finite or infinite measure), and provide pointwise description of them.
Let (e_i) be a dictionary for a separable Banach space X. We consider the problem of approximation by linear combinations of dictionary elements with quantized coefficients drawn usually from a `finite alphabet'. We investigate several approximation properties of this type and connect them to the Banach space geometry of X. The existence of a total minimal system with one of these properties, namely the coefficient quantization property, is shown to be equivalent to X containing c_0.
We prove several results concerning the representation of projections on arbitrary Banach spaces. We also give illustrative examples including an example of a generalized bi-circular projection which can not be written as the average of the identity with an isometric reflection. We also characterize generalized bi-circular projections on $C_0(\Om,X)$, with $\Om$ a locally compact Hausdorff space (not necessarily connected) and $X$ a Banach space with trivial centralizer.
We prove that every Banach space, not necessarily separable, can be isometrically embedded into a $\mathcal L_{\infty}$-space in a way that the corresponding quotient has the Radon-Nikodym and the Schur properties. As a consequence, we obtain $\mathcal L_\infty$ spaces of arbitrary large densities with the Schur and the Radon-Nikodym properties. This extents the a classical result by J. Bourgain and G. Pisier.
We show that a complemented subspace of a locally convex direct sum of an uncountable collection of Banach spaces is a locally convex direct sum of complemented subspaces of countable subsums. As a corollary we prove that a complemented subspace of a locally convex direct sum of arbitrary collection of $\ell_{1}(Γ)$-spaces is isomorphic to a locally convex direct sum of $\ell_{1}(Γ)$-spaces.
Selections of integrable multifunctions in arbitrary Banach spaces
该研究在任意（不必可分的）Banach 空间中考察多值函数的可积选择存在性，并用以将多值函数表示为性质更好的多值函数之平移。
We continue the investigation of the behaviour of octahedral norms in tensor products of Banach spaces. Firstly, we will prove the existence of a Banach space $Y$ such that the injective tensor products $l_1\widehat{\otimes}_\varepsilon Y$ and $L_1\widehat{\otimes}_\varepsilon Y$ both fail to have an octahedral norm, which solves two open problems from the literature. Secondly, we will show that in the presence of the metric approximation property octahedrality is preserved from a non-reflexive $L$-embedded Banach space taking projective tensor products with an arbitrary Banach space.
In order to measure qualitative properties we introduce a notion of a type for arbitrary normed spaces which measures the worst possible growth of partial sums of sequences weakly converging to zero. The ideas can be traced back to Banach and Mazur who used this type to compare the so-called linear dimension of classical Banach spaces. As an application we compare the linear dimension and investigate isomorphy of some classical Banach spaces.
We study the regularity properties of the inverse of a bilipschitz mapping $f$ belonging $W^m X_{\text{loc}}$, where $X$ is an arbitrary Banach function space. Namely, we prove that the inverse mapping $f^{-1}$ is also in $W^m X_{\text{loc}}$. Furthermore, the paper shows that the class of bilipschitz mappings in $W^m X_{\text{loc}}$ is closed with respect to composition and multiplication.
In this paper, we study the notion of $ϕ$-injectivity in the special case that $ϕ=0$. For an arbitrary locally compact group $G$, we characterize the 0-injectivity of $L^{1}(G)$ as a left $L^{1}(G)$ module. Also, we show that $L^{1}(G)^{**}$ and $L^{p}(G)$ for $1<p<\infty$ are 0-injective Banach $L^{1}(G)$ modules.
This paper studies Schauder frames in Banach spaces, a concept which is a natural generalization of frames in Hilbert spaces and Schauder bases in Banach spaces. The associated minimal and maximal spaces are introduced, as are shrinking and boundedly complete Schauder frames. Our main results extend the classical duality theorems on bases to the situation of Schauder frames. In particular, we will generalize James' results on shrinking and boundedly complete bases to frames. Secondly we will extend his characterization of the reflexivity of spaces with unconditional bases to spaces with uncond…
This work performs a study of the category of complete matrix-normed spaces, called matricial Banach spaces. Many of the usual constructions of Banach spaces extend in a natural way to matricial Banach spaces, including products, direct sums, and completions. Also, while the minimal matrix-norm on a Banach space is well-known, this work characterizes the maximal matrix-norm on a Banach space from the work of Effros and Ruan as a dual operator space. Moreover, building from the work of Blecher, Ruan, and Sinclair, the Haagerup tensor product is merged with the direct sum to form a Haagerup tens…
We construct a class of minimal trees and use these trees to establish a number of coloring theorems on general trees. Among the applications of these trees and coloring theorems are quantification of the Bourgain $\ell_p$ and $c_0$ indices, dualization of the Bourgain $c_0$ index, establishing sharp positive and negative results for constant reduction, and estimating the Bourgain $\ell_p$ index of an arbitrary Banach space $X$ in terms of a subspace $Y$ and the quotient $X/Y$.
We obtain Lebesgue-type inequalities for the greedy algorithm for arbitrary complete seminormalized biorthogonal systems in Banach spaces. The bounds are given only in terms of the upper democracy functions of the basis and its dual. We also show that these estimates are equivalent to embeddings between the given Banach space and certain discrete weighted Lorentz spaces. Finally, the asymptotic optimality of these inequalities is illustrated in various examples of non necessarily quasi-greedy bases.
We provide some new consequences on the Lipschitz numerical radius and index which were introduced recently. More precisely, we give some renorming results on the Lipschitz numerical index, introduce a concept of Lipschitz numerical radius attaining functions in order to show that the denseness fails for an arbitrary Banach space, and study a Lipschitz version of Daugavet centers. Furthermore, we discuss the Lipschitz numerical index of vector-valued function spaces, absolute sums of Banach spaces, the Köthe-Bochner spaces, and Banach spaces which contain a dense union of increasing family of …
Sets with arbitrary Hausdorff and packing scales in infinite dimensional Banach spaces
本文证明对满足温和假设的任意Hausdorff函数对，可在Baire空间中构造紧集使其Hausdorff测度与packing测度均有限且正，并嵌入任意无限维Banach空间，肯定回答了Fan关于任意尺度度量空间存在性的问题。
We show that any Banach space contains a continuum of non isomorphic subspaces or a minimal subspace. We define an ergodic Banach space $X$ as a space such that $E_0$ Borel reduces to isomorphism on the set of subspaces of $X$, and show that every Banach space is either ergodic or contains a subspace with an unconditional basis $ which is complementably universal for the family of its block-subspaces. We also use our methods to get uniformity results; for example, in combination with a result of B. Maurey, V. Milman and N. Tomczak-Jaegermann, we show that an unconditional basis of a Banach spa…
We answer the question of W.T. Gowers, giving an example of a bounded operator on a subspace of Gowers unconditional space which is not a strictly singular perturbation of a restriction of a diagonal operator. We make some observations on operators in arbitrary tight by support Banach space, showing in particular that in such space no two isomorphic infinitely dimensional subspaces form a direct sum.
In finitely-dimensional spaces the sum range of a series has to be an affine subspace. It is long known this is not the case in infinitely dimensional Banach spaces. In particular in 1984 M.I. Kadets and K. Wozniakowski obtained an example of a series the sum range of which consisted of two points, and asked whether it is possible to obtain more than two, but finitely many points. This paper answers the question positively, by showing how to obtain an arbitrary finite set as the sum range of a series in any infinitely dimensional Banach space.
It is known that if $M$ is a finite-dimensional Banach space, or a strictly convex space, or the space $\ell_1$, then every non-expansive bijection $F: B_M \to B_M$ is an isometry. We extend these results to non-expansive bijections $F: B_E \to B_M$ between unit balls of two different Banach spaces. Namely, if $E$ is an arbitrary Banach space and $M$ is finite-dimensional or strictly convex, or the space $\ell_1$ then every non-expansive bijection $F: B_E \to B_M$ is an isometry.
An L-embedded Banach spaace is a Banach space which is complemented in its bidual such that the norm is additive between the two complementary parts. On such spaces we define a topology, called an abstract measure topology, which by known results coincides with the usual measure topology on preduals of finite von Neumann algebras (like $L_1([0,1])$). Though not numerous, the known properties of this topology suffice to generalize several results on subspaces of $L_1([0,1])$ to subspaces of arbitrary L-embedded spaces.
We extend to Banach space nest algebras the theory of essential supports and support function pairs of their bimodules, thereby obtaining Banach space counterparts of long established results for Hilbert space nest algebras. Namely, given a Banach space nest algebra $\mathcal A$, we charaterise the maximal and the minimal $\mathcal A$-bimodules having a given essential support function or support function pair. These characterisations are complete except for the minimal $\mathcal A$-bimodule corresponding to a support function pair, in which case we make some headway. We also show that the wea…
We extend Feichtinger's minimality property on smallest non-trivial time-frequency shift invariant Banach spaces, to the quasi-Banach case. Analogous properties are deduced for certain matrix classes. We use these results to prove that $\operatorname{Op}(a)$ is a Schatten-$q$ operator from $M^\infty$ to $M^p$ and $r$-nuclear operator from $M^\infty$ to $M^r$ when $a\in M^r$ for suitable $p$, $q$ and $r$ in $(0,\infty ]$.
The main result of this paper establishes that the known Arazy-Cwikel property holds for classes of uniformly K-monotone spaces in the quasi-Banach setting provided that the initial couple is mutually closed. As a consequence, we get that the class of all quasi-Banach K-spaces (i.e., interpolation spaces which are described by the real K-method) with respect to an arbitrary mutually closed Banach couple enjoys the Arazy-Cwikel property. Another consequence complements some previous results by Bykov and Ovchinnikov, showing that this property holds also for the class of all interpolation quasi-…
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。