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Let X be a complex Banach space, and denote by \mathcal {B}(X) the algebra of all bounded linear operators on X. Let C,D\in \mathcal {B} \left (X\right) be fixed operators. In this paper, we characterize linear, continuous and bijective maps \varphi and \psi on \mathcal {B}\left (X\right) for which there exist invertible operators T_0, W_0 \in \mathcal { B}(X) such that \varphi (T_0), \psi (W_0) \in \mathcal {B}(X) are both invertible, having the property that \varphi \left (A\right) \psi \left (B\right) =D in \mathcal {B}(X) whenever AB=C in \mathcal {B}(X). As a corollary, we deduce the form of linear, bijective and continuous maps \varphi on \mathcal {B}(X) having the property that \varphi \left (A\right) \varphi \left (B\right) =D in \mathcal {B}(X) whenever AB=C. [ABSTRACT FROM AUTHOR]

Characterization of simultaneous similarity for commuting m- tuples of operators is an open problem even in finite-dimensional spaces; known as "A wild problem in linear algebra". In this paper we offer a criterion for simultaneous similarity of m-tuples of k-cyclic commuting operators on an arbitrary Banach space. Moreover, we obtain an additional equivalence condition in the case of finite dimensional Banach spaces, which extends the result found by Shekhtman [Math. Stat. 1 (2013), pp. 157–161] for pairs of cyclic commuting matrices. We also present two applications of our results, one in the case of general multiplication operators on Banach spaces of analytic function, and one for m-tuples of commuting square matrices. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form D^{\alpha }_Cu(t)=Au(t)+f(t), u(0)=x, 0<\alpha \le 1, (*) where D^{\alpha }_Cu(t) is the derivative of the function u in the Caputo's sense, A is a linear operator in a Banach space \mathbb {X} that may be unbounded and f satisfies the property that \lim _{t\to \infty } (f(t+1)-f(t))=0 which we will call asymptotic 1-periodicity. By using the spectral theory of functions on the half line we derive analogs of Katznelson-Tzafriri and Massera Theorems. Namely, we give sufficient conditions in terms of spectral properties of the operator A for all asymptotic mild solutions of Eq. (*) to be asymptotic 1-periodic, or there exists an asymptotic mild solution that is asymptotic 1-periodic. [ABSTRACT FROM AUTHOR]

This paper considers L^1-spaces of covariant functions of characters of normal subgroups. Assume that G is a locally compact group with the group algebra L^1(G) and N is a closed and normal subgroup of G. Suppose that \mathbb {T} is the circle group and \xi :N\to \mathbb {T} is a character. We introduce the Banach space L_\xi ^1(G,N) of covariant functions of \xi on G and present an operator theoretic approach to study structure of L^1_\xi (G,N). It is proven that L^1_\xi (G,N) and a quotient space of L^1(G) are isometrically isomorphic. The paper is concluded by constructive characterizations for covariant functions of \xi. [ABSTRACT FROM AUTHOR]

This paper deals with the chaotic behavior of linear operators on Banach spaces in both discrete and continuous cases. The inheritances of chaos for linear operators and C_0-semigroups are obtained. More precisely, for any positive integer n\geq 2, both Li-Yorke n-chaos and distributional n-chaos are proved to be inherited under iterations for a linear operator. One further shows that a C_0-semigroup \{T_t\}_{t\geq 0} and every single operator T_t share the same Li-Yorke n-scrambled set and distributionally n-scrambled set for any positive integer n\geq 2. In particular, the Li-Yorke n-chaos and distributional n-chaos become the Li-Yorke chaos and distributional chaos when n=2, respectively. Some equivalent criteria for dense n-chaos and generic n-chaos of linear operators and C_0-semigroups are also established for any n\geq 2. [ABSTRACT FROM AUTHOR]

We show that well-established methods from the theory of Banach modules and time-frequency analysis allow to derive completeness results for the collection of shifted and dilated version of a given (test) function in a quite general setting. While the basic ideas show strong similarity to the arguments used in a recent paper by V. Katsnelson we extend his results in several directions, both relaxing the assumptions and widening the range of applications. There is no need for the Banach spaces considered to be embedded into (L2(R), ||⋅||2), nor is the Hilbert space structure relevant. We choose to present the results in the setting of the Euclidean spaces, because then the Schwartz space S'(Rd) (d ≥ 1) of tempered distributions provides a well-established environment for mathematical analysis. We also establish connections to modulation spaces and Shubin classes (Qs(Rd), ||⋅||Qs), showing that they are special cases of Katsnelson's setting (only) for s ≥ 0. [ABSTRACT FROM AUTHOR]

In this paper, we explore relatively quasimöbius invariance of \varphi-uniform domains and natural domains. Firstly, we prove that the control function of relatively quasimöbius mappings can be chosen in a power form. Applying this observation and a deformed cross–ratio introduced by Bonk and Kleiner, we next show that relatively quasimöbius mappings are coarsely bilipschitz in the distance ratio metric. Combined with the assumption that the mapping is coarsely bilipschitz in the quasihyperbolic metric, we establish relatively quasimöbius invariance of \varphi-uniform domains and natural domains. As a by-product, we also obtain a similar result for uniform domains which provides a new method to answer a question posed by Väisälä. [ABSTRACT FROM AUTHOR]

For two Banach spaces X and Y, we denote by K(X,Y) (resp. B(X,Y)) the space of all compact (resp. bounded) linear operators from X to Y. In this paper, we show that for 1\leq p,q<\infty, K(\ell _{p},\ell _{q}) is an 8-Lipschitz retract of B(\ell _{p},\ell _{q}). [ABSTRACT FROM AUTHOR]

We provide, in every infinite-dimensional separable Banach space, an average locally uniformly rotund (and hence rotund) Gâteaux smooth renorming which is not locally uniformly rotund. This solves an open problem posed in a recent monograph by A.J. Guirao, V. Montesinos, and V. Zizler. [ABSTRACT FROM AUTHOR]

This paper deals with the dynamics of a system governed by a multi-dimensional linear hyperbolic PDE. The dynamical behaviors of linear PDEs extremely depend on the selection of space, and a conventional way is to define a infinite-dimensional space with a tuning parameter. Thereby, the linear PDEs can exhibit chaos or stability in the different range of tuning parameter. In this work, the chaos of the C_0-semigroup corresponding to the system is established on the Banach space of multivariate analytic functions when the tuning parameter exceeds some given positive number. Based on this, both Devaney and distributional chaos of the system are further obtained. Meanwhile, the C_0-semigroup is proved to be uniformly exponentially stable when the tuning parameter is less than a certain positive number, which contributes to showing the global stability of the system. Finally, two examples are given to illustrate effectiveness of our results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce sequentially defined Besov spaces adapted to the scaling invariant property of the 3D incompressible Navier-Stokes equations. We first show that these spaces are Banach spaces. We then establish regularity conditions in these spaces. [ABSTRACT FROM AUTHOR]

We provide an elementary proof of a result by V. P. Fonf and C. Zanco on point-finite coverings of separable Hilbert spaces. Indeed, by using a variation of the famous argument introduced by J. Lindenstrauss and R. R. Phelps [Israel J. Math. 6 (1968), pp. 39–48] to prove that the unit ball of a reflexive infinite-dimensional Banach space has uncountably many extreme points, we prove the following result. Let X be an infinite-dimensional Hilbert space satisfying dens(X)< 2ℵ0, then X does not admit point-finite coverings by open or closed balls, each of positive radius. In the second part of the paper, we follow the argument introduced by V. P. Fonf, M. Levin, and C. Zanco in [J. Geom. Anal. 24 (2014), pp. 1891–1897] to prove that the previous result holds also in infinite-dimensional Banach spaces that are both uniformly rotund and uniformly smooth. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of periodic solutions and attractiveness for some partial functional differential equations in Banach spaces. We assume that the first linear part generates a strongly continuous semigroup, while the delayed part is periodic with respect to the first argument. We prove that the existence of a bounded solution implies the existence of a periodic solution. Several results regarding uniqueness and global attractiveness of periodic solutions are also established. The analysis relies on a fixed point theorem of Chow and Hale's type and uses some arguments of weak topology. Our theorems extend in a broad sense some new and classical related results. An application to a transport equation with delay is also presented. [ABSTRACT FROM AUTHOR]

Let π : G → B(E) be a continuous representation of a compact group G on a Banach space E. We prove that the set of vectors π (h)x, as h runs through the set T(G) of all trigonometric polynomials on G, and x runs through E, spans an invariant dense linear subspace of E. We prove the existence of a topological direct sum decomposition E = ⊕θ ∈ GEθ for E, where each Eθ is a closed π-invariant subspace of E. If λp : M(G) → B(Lp(G)), p ∈(1,∞), is the left regular representation of the measure algebra M(G) and B ⊂ PMp(G) is a homogeneous Banach space, we show that B λp(T(G)) is norm dense in B. Since Hilbert space techniques are not available, new machinery is developed in the paper for the proofs. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the study of a long-standing open problem posed by Lau in 1976. This problem is about whether the left amenability property of the space of left uniformly continuous functions of a semitopological semigroup is equivalent to the existence of a common fixed point for every jointly weak* continuous norm nonexpansive action on a nonempty weak* compact convex subset of a dual Banach space. We establish in this paper a positive answer. [ABSTRACT FROM AUTHOR]

A remarkable result of Molnár [Proc. Amer. Math. Soc. 126 (1998), pp. 853–861] states that automorphisms of the algebra of operators acting on a separable Hilbert space are stable under "small" perturbations. More precisely, if \phi,\psi are endomorphisms of \mathcal {B}(\mathcal {H}) such that \|\phi (A)-\psi (A)\|<\|A\| and \psi is surjective, then so is \phi. The aim of this paper is to extend this result to a larger class of Banach spaces including \ell _p and L_p spaces, where 1

Suppose that K is a compact Hausdorff space, S is a locally compact Hausdorff space and X is a Banach space with Schäffer constant \mathrm {S}(X). In this paper, we prove that if there is a map T from C(K) to C_{0}(S, X) satisfying \begin{equation*} \frac {1}{M} \|f-g\| \leq \|T(f)-T(g)\|\leq M \|f-g\|,\ \forall f, g \in C(K), \end{equation*} with 1\leq M^{2}<\mathrm {S}(X), then there exists a compact subset S_{0} of S and a continuous function \varphi from S_{0} onto K. This theorem on Lipschitz embeddings of C(K) into C_{0}(S, X) is the first nonlinear vector extension of the classical 1966 Holsztyński Theorem. Our result is optimal for many Banach spaces X including the spaces \ell _p^n, \ell _p and L_{p}([0,1]), 1 < p <\infty, n \geq 2, even when T is linear. The motivation to prove this result comes from the fact that it immediately yields a nontrivial lower bound for the C_{0}(S, X)-distortion of the class of all separable Banach spaces whenever S is a scattered space and \mathrm {S}(X)>1, namely \mathrm {S}(X) itself. [ABSTRACT FROM AUTHOR]

The \lambda-property and the uniform λ-property were first introduced by R. Aron and R. Lohman in 1987 as geometric properties of Banach spaces. In 1989, Th. Shura and D. Trautman showed that the Schreier space possesses the λ-property and asked if it has the uniform λ-property. In this paper, we show that Schreier space does not have the uniform λ-property. Furthermore, we show that the dual of the Schreier space does not have the uniform λ-property. [ABSTRACT FROM AUTHOR]

This paper is devoted to exponential dichotomies of nonautonomous difference equations. Under the assumptions that (Am)m∈Z is a sequence of bounded operators acting on an arbitrary Banach space X that admits a uniform exponential dichotomy and that (Bm)m∈Z is a sequence of compact operators such that lim|m|→∞||Bm|| = 0, D. Henry proved that either the sequence (Am+Bm)m∈Z admits a uniform exponential dichotomy or there exists a bounded nonzero sequence (xm)m∈Z ⊂ X such that xm+1 = (Am +Bm)xm for each m ∈ Z. In this paper we prove Henry's result in the setting of nonuniform exponential dichotomies. Then we obtain a result on roughness of the nonuniform exponential dichotomy and give stability of Lyapunov exponents. In addition, we establish corresponding results for dynamics with continuous time. [ABSTRACT FROM AUTHOR]

In this paper, we continue our study of the quasihyperbolic metric over Banach spaces. The main results of the paper present a criterion for smoothness of geodesics of quasihyperbolic type metrics in Banach spaces, under a Dini type condition on the weight function, which improves an earlier result of the first two authors. We also answer a question posed by the first two authors in an earlier paper with R. Klén and present results related to the question of smoothness of quasihyperbolic balls. [ABSTRACT FROM AUTHOR]

When Journé established the classical multi-parameter singular integral theory in 1985, it was formulated in the language of vector-valued Calderón-Zygmund theory. In 2011, Pott and Villarroya formulated a new T1 Theorem for product spaces. In their work, they replaced the vectorvalued formulations for several mixed type conditions. Inspired by their work, Martikainen redefined the bi-parameter operators. Originally, it was thought that this new class of operators was larger (and contained the ones defined by Journé). In this paper we prove that both classes are actually equal. [ABSTRACT FROM AUTHOR]

The classical Schur theorem asserts that the weak convergence and the norm convergence in the Banach space l1 coincide. In this paper we study complexity and cardinality of subfamilies F of ℘(ω) such that a sequence 〈xn : n ∈ ω〉 ⊆ l1 is norm convergent whenever limn→∞∑j∈A xn(j) = 0 for every A ∈ F. We call such families Schur and prove that they cannot have cardinality less than the pseudo-intersection number p. On the other hand, we also show that every non-meager subset of the Cantor space 2ω is a Schur family when thought of as a subset of ℘(ω), implying that the minimal size of a Schur family is bounded from above by non(M), the uniformity number of the ideal of meager subsets of 2ω. [ABSTRACT FROM AUTHOR]

Diamond graphs and binary trees are important examples in the theory of metric embeddings and also in the theory of metric characterizations of Banach spaces. Some results for these families of graphs are parallel to each other; for example superreflexivity of Banach spaces can be characterized both in terms of binary trees (Bourgain, 1986) and diamond graphs (Johnson-Schechtman, 2009). In this connection, it is natural to ask whether one of these families admits uniformly bilipschitz embeddings into the other. This question was answered in the negative by Ostrovskii (2014), who left it open to determine the order of growth of the distortions. The main purpose of this paper is to get a sharp up-to-a-logarithmic-factor estimate for the distortions of embeddings of binary trees into diamond graphs and, more generally, into diamond graphs of any finite branching k ≥ 2. Estimates for distortions of embeddings of diamonds into infinitely branching diamonds are also obtained. [ABSTRACT FROM AUTHOR]

Kadets and Lyubich have shown that a weakly almost periodic representation of a topological group on a Banach space has the property that a vector with almost periodic coefficients is almost periodic. In this paper we obtain the analogous result for distal representations of semigroups on quasi-complete locally convex spaces, removing the condition that the representation be weakly almost periodic and requiring only that the coefficients of the representation be weakly almost periodic. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the sharp conditions of the inclusion relations between modulation spaces Ms p,q and Triebel-Lizorkin spaces Fp,r for p ≤ 1, which greatly improve and extend the results for the embedding relations between local Hardy spaces and modulation spaces obtained by Kobayashi, Miyachi and Tomita in [Studia Math. 192 (2009), 79-96]. [ABSTRACT FROM AUTHOR]

This article provides a relation between the classical Bohr radius, the arithmetic Bohr radius, and the multidimensional Bohr radius. Through this relation, we give answers to some questions, one is raised by Defant et al. and other one is an open problem, both are related to the multidimensional Bohr radius. Moreover, we provide better new estimates of Bohr radii for holomorphic functions defined on Banach spaces. [ABSTRACT FROM AUTHOR]

Let X, Y be Banach spaces, and let ƒ : X → Y be an ε-isometry with ƒ(0) = 0 for some ε ≥ 0. In this paper, we show that for every x* ∈ X*, there exists φ ∈ Y* with ||φ|| = ||x*|| ≡ r such that || − <φ,ƒ(x)>| ≤ 3rε, ∨x ∈ X. Moreover, the constant ''3'' is the best one in general. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the relatively compact sets in quasi-Banach function spaces, providing an important improvement of the known results. As an application, we take the final step in establishing a relative compactness criteria for function spaces with any weight without any assumption. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new idea of studying the weighted pseudo almost periodic phenomenon. We introduce a new mathematical concept of α ~ l relatively dense set and show that the α ~ l relatively dense set is more useful than the classical relatively dense set due to Bohr or Bochner in treating the weighted pseudo almost periodic phenomenon. We prove the uniqueness of decomposition of the weighted pseudo almost periodic functions with values in Banach spaces and with weights in a new weight function set, and show that the new weight set is larger than all the existing weight function sets which ensures the uniqueness of such decomposition in literature. Moreover, we reveal several properties of the vector-valued weighted pseudo almost periodic functions, and provide a series of concrete examples and remarks for illustrating our theoretical results. [ABSTRACT FROM AUTHOR]

In 1976 during a conference in Halifax, A. Lau raised the question on whether left amenability property for a semitopological semigroup is sufficient to ensure the existence of a common fixed point for every jointly weak* continuous norm nonexpansive action on a nonempty weak* compact convex set in a dual Banach space. In this paper we establish a positive answer for discrete semigroups and for strongly left amenable semitopological semigroups. [ABSTRACT FROM AUTHOR]

A well-known result in [Hsu-Smith-Waltman, Trans. Amer. Math. Soc. (1996)] states that in a competitive semiflow defined on X+ = X+1 × X+2, the product of two cones in respective Banach spaces, if (u*, 0) and (0,v*) are the global attractors in X+1 × {0} and {0} × X+2 respectively, then one of the following three outcomes is possible for the two competitors: either there is at least one coexistence steady state, or one of (u*, 0), (0,v*) attracts all trajectories initiating in the order interval I = [0, u*] × [0, v*]. However, it was demonstrated by an example that in some cases neither (u*,0) nor (0,v*) is globally asymptotically stable if we broaden our scope to all of X+. In this paper, we give two sufficient conditions that guarantee, in the absence of coexistence steady states, the global asymptotic stability of one of (u*, 0) or (0, v*) among all trajectories in X+. Namely, one of (u*,0) or (0,v*) is (i) linearly unstable, or (ii) linearly neutrally stable but zero is a simple eigenvalue. Our results complement the counterexample mentioned in the above paper as well as applications that frequently arise in practice. [ABSTRACT FROM AUTHOR]

Let X be a completely regular Hausdorff space, let E and F denote Banach spaces. Let C_b(X,E) denote the space of E-valued bounded continuous functions on X and let \beta be the strict topology on this space. We establish the relationship between nuclear operators T:C_b(X,E)\rightarrow F between the locally convex space (C_b(X,E),\beta) and the Banach space F and their representing operator-valued Borel measures. [ABSTRACT FROM AUTHOR]

The purpose in this article is to discuss under what conditions a continuous pseudocontractive mapping, in particular a nonexpansive map, has a fixed point when its domain is a closed, convex and unbounded subset of a Banach space. In fact, we will give a characterization of this property. The Halpern iterative algorithm for nonexpansive mappings is also investigated. [ABSTRACT FROM AUTHOR]

We establish a general framework for representability of a metric group on a (well-behaved) class of Banach spaces. More precisely, let \mathcal {G} be a topological group, and \mathcal {A} a unital symmetric C^*-subalgebra of \mathrm {UC}(\mathcal {G}), the algebra of bounded uniformly continuous functions on \mathcal {G}. Generalizing the notion of a stable metric, we study \mathcal {A}-metrics \delta, i.e., the function \delta (e, \cdot) belongs to \mathcal {A}; the case \mathcal {A}=W\hskip -0.7mm A\hskip -0.2mm P(\mathcal {G}), the algebra of weakly almost periodic functions on \mathcal {G}, recovers stability. If the topology of G is induced by a left invariant metric d, we prove that \mathcal {A} determines the topology of \mathcal {G} if and only if d is uniformly equivalent to a left invariant \mathcal {A}-metric. As an application, we show that the additive group of C[0,1] is not reflexively representable; this is a new proof of Megrelishvili [ Topological transformation groups: selected topics , Elsevier, 2007, Question 6.7] (the problem was already solved by Ferri and Galindo [Studia Math. 193 (2009), pp. 99–108] with different methods and later the results were generalized by Yaacov, Berenstein, and Ferri [Math. Z. 267 (2011), pp.129–138]). Let now \mathcal {G} be a metric group, and assume \mathcal {A}\subseteq \mathrm {LUC}(\mathcal {G}), the algebra of bounded left uniformly continuous functions on \mathcal {G}, is a unital C^*-algebra which is the uniform closure of coefficients of representations of \mathcal {G} on members of \mathscr {F}, where \mathscr {F} is a class of Banach spaces closed under \ell _2-direct sums. We prove that \mathcal {A} determines the topology of \mathcal {G} if and only if \mathcal {G} embeds into the isometry group of a member of \mathscr {F}, equipped with the weak operator topology. As applications, we obtain characterizations of unitary and reflexive representability. [ABSTRACT FROM AUTHOR]

Akemann and Weaver (2014) have shown a remarkable extension of Weaver's KSr Conjecture (2004) in the form of approximate Lyapunov's theorem. This was made possible thanks to the breakthrough solution of the Kadison-Singer problem by Marcus, Spielman, and Srivastava (2015). In this paper we show a similar type of Lyapunov's theorem for continuous frames on non-atomic measure spaces. In contrast with discrete frames, the proof of this result does not rely on the recent solution of the Kadison-Singer problem. [ABSTRACT FROM AUTHOR]

The purpose of the present work is to answer an open problem which was raised by G. Costakis and A. Manoussos in their paper "J-class operators and hypercyclicity", J. Operator Theory 67 (2012), 101-119. More precisely, we give the spectral description of the closure of the set of J-class operators acting on a separable Hilbert space. [ABSTRACT FROM AUTHOR]

If X is a compact subset of a Banach space with X -X homogeneous (equivalently 'doubling' or with finite Assouad dimension), then X can be embedded into some R...n (with n sufficiently large) using a linear map L whose inverse is Lipschitz to within logarithmic corrections. More precisely, there exist c, α > 0 such that c ‖x - y‖/| log ‖x - y‖ |α ≤ |Lx - Ly| ≤ c‖x - y‖ for all x, y ∊ X, ‖x - y‖ < δ, for some δ sufficiently small. It is known that one must have α > 1 in the case of a general Banach space and α > 1/2 in the case of a Hilbert space. It is shown in this paper that these exponents can be achieved. While the argument in a general Banach space is relatively straightforward, the Hilbert space case relies on the fact that the maximum volume of a hyperplane slice of a k-fold product of unit volume N-balls is bounded independent of k (this provides a 'qualitative' generalisation of a result on slices of the unit cube due to Hensley (Proc.AMS 73 (1979), 95-100) and Ball (Proc.AMS 97 (1986), 465-473)). [ABSTRACT FROM AUTHOR]

In this paper, a nonautonomous neutral delay prey-predator model with impulse and Hassell-Varley type functional response is studied. By using the continuation theorem of coincidence degree theory, easily verifiable criteria are established for the existence of positive periodic solutions to the system. [ABSTRACT FROM AUTHOR]

Let \Sigma be a \sigma-algebra of subsets of a set \Omega and X be a Banach space, and let B(\Sigma,X) stand for the Banach space of all X-valued totally \Sigma-measurable functions on \Omega, equipped with the sup-norm. We study nuclear operators T:B(\Sigma,X)\rightarrow Y between Banach spaces B(\Sigma,X) and Y in terms of their representing measures m:\Sigma \rightarrow \mathcal {N}(X,Y), where \mathcal {N}(X,Y) stands for the Banach space of all nuclear operators U:X\rightarrow Y, equipped with the nuclear norm. We establish the relationship between nuclearity of a bounded linear operator T:B(\Sigma,X)\rightarrow Y and nuclearity of its conjugate operator. [ABSTRACT FROM AUTHOR]

There is a long-standing problem, posed by A.T.-M. Lau [ Fixed point theory and its applications , Academic Press, New York-London, 1976, pp. 121–129], whether left amenability is sufficient to ensure the existence of a common fixed point for every jointly weak^{\ast } continuous nonexpansive semigroup action on a nonempty weak^{\ast } compact convex set in a dual Banach space. In this note we discuss the current status of this problem and give a partial solution in the case of weak^{\ast } compact convex sets with the Radon–Nikodým property. [ABSTRACT FROM AUTHOR]

Using elementary probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, we prove that for every infinite compact spaces K and L the product K\times L admits a sequence \langle \mu _n\colon n\in \mathbb {N}\rangle of normalized signed measures with finite supports which converges to 0 with respect to the weak* topology of the dual Banach space C(K\times L)^*. Our approach is completely constructive—the measures \mu _n are defined by an explicit simple formula. We also show that this result generalizes the classical theorem of Cembranos [Proc. Amer. Math. Soc. 91 (1984), pp. 556–558] and Freniche [Math. Ann. 267 (1984), pp. 479–486] which states that for every infinite compact spaces K and L the Banach space C(K\times L) contains a complemented copy of the space c_0. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with linear control systems on time scales. We show that, under appropriate hypotheses, the self-accessible trajectories have diameter greater than or equal to a certain fixed positive number. [ABSTRACT FROM AUTHOR]

Let Y be a sublattice of a vector lattice X. We consider the problem of identifying the smallest order closed sublattice of X containing Y. It is known that the analogy with topological closure fails. Let ... be the order closure of Y consisting of all order limits of nets of elements from Y. Then ... need not be order closed. We show that in many cases the smallest order closed sublattice containing Y is in fact the second order closure .... Moreover, if X is a σ-order complete Banach lattice, then the condition that ... is order closed for every sublattice Y characterizes order continuity of the norm of X. The present paper provides a general approach to a fundamental result in financial economics concerning the spanning power of options written on a financial asset. [ABSTRACT FROM AUTHOR]

In this paper, we study the compatibility between Talagrand's inequality and the concentration topology; i.e., if a sequence of mm-spaces satisfying Talagrand's inequality converges with respect to the observable distance, then the limit space satisfies Talagrand's inequality. [ABSTRACT FROM AUTHOR]

We study Hilbert generated versions of nonseparable Banach spaces \mathcal {X} considered by Shelah, Steprāns and Wark where the behavior of the norm on nonseparable subsets is so irregular that it does not allow any linear bounded operator on \mathcal {X} other than a diagonal operator (or a scalar multiple of the identity) plus a separable range operator. We address the questions as to whether these spaces admit uncountable equilateral sets and whether their unit spheres admit uncountable (1+)-separated or (1+\varepsilon)-separated sets. We resolve some of the above questions for two types of these spaces by showing both absolute and undecidability results. The corollaries are that the continuum hypothesis (in fact: the existence of a nonmeager set of reals of the first uncountable cardinality) implies the existence of an equivalent renorming of the nonseparable Hilbert space \ell _2(\omega _1) which does not admit any uncountable equilateral set and it implies the existence of a nonseparable Hilbert generated Banach space containing an isomorphic copy of \ell _2 in each nonseparable subspace, whose unit sphere does not admit an uncountable equilateral set and does not admit an uncountable (1+\varepsilon)-separated set for any \varepsilon >0. This could be compared with a recent result by Hájek, Kania and Russo saying that all nonseparable reflexive Banach spaces admit uncountable (1+\varepsilon)-separated sets in their unit spheres. [ABSTRACT FROM AUTHOR]

We exhibit a Banach space Z failing the approximation property, for which there is an uncountable family \mathscr F of closed subideals contained in the Banach algebra \mathcal K(Z) of the compact operators on Z, such that the subideals in \mathscr F are mutually isomorphic as Banach algebras. This contrasts with the behaviour of closed ideals of the algebras \mathcal L(X) of bounded operators on X, where closed ideals \mathcal I \neq \mathcal J are never isomorphic as Banach algebras. We also construct families of non-trivial closed subideals contained in the strictly singular operators \mathcal S(X) for classical spaces such as X = L^p with p \neq 2, where pairwise isomorphic as well as pairwise non-isomorphic subideals occur. [ABSTRACT FROM AUTHOR]

We show that the predual of a JBW*-triple has the weak Banach- Saks property, that is, reflexive subspaces of a JBW*-triple predual are superreflexive. We also prove that JBW*-triple preduals satisfy the Koml'os property (which can be considered an abstract version of the weak law of large numbers). The results rely on two previous papers from which we infer the fact that, like in the classical case of L¹, a subspace of a JBW*-triple predual contains 1 as soon as it contains uniform copies of ℓ1n. [ABSTRACT FROM AUTHOR]

It was proved by Dodos and Ferenczi that the classes of Banach spaces with a separable dual and of separable reflexive Banach spaces are strongly bounded. In this paper, we provide an isometric version of this result. [ABSTRACT FROM AUTHOR]

We consider quasi-isometries in real continuous functions spaces and show that such a quasi-isometry can be well approximated by an affine surjective isometry. On the other hand, we give an example of quasi-isometries of the unit ball B_H in a Hilbert space H that are far from any affine map of H and from any isometry of B_H. [ABSTRACT FROM AUTHOR]

We show that there exists a Banach space X which contains closed subspaces Y and Z with Y+Z=X such that the associated surjective summation operator \Sigma \colon Y\times Z\to X defined by \Sigma (y,z)= y+z for y\in Y and z\in Z has no Lipschitz right inverse. [ABSTRACT FROM AUTHOR]