In this paper, we give simplified and equivalent characterizations of Banach limit functional, which is the minimum requirement to characterize strong almost convergence. With this machinery, we show that Hajdukovic's quasi-almost convergence is equivalent to strong almost convergence.
We prove an Ulam type stability result for a non-associative version of the multiplicativity equation, that is, $T(xy)=Ψ(T(x),T(y))$, where $T$ is a unital bounded operator acting from an amenable Banach algebra to a dual Banach algebra.
We study almost square Banach spaces under a topological point of view. Indeed, we prove that the class of Banach spaces which admits an equivalent norm to be ASQ is that of those Banach spaces which contain an isomorphic copy of $c_0$. We also prove that the symmetric projective tensor products of an almost square Banach space have the strong diameter two property
The aim of this note is to study some geometrical properties like diameter two properties, octahedrality and almost squareness in the setting of (symmetric) tensor product spaces. In particular, we show that the injective tensor product of two octahedral Banach spaces is always octahedral, the injective tensor product of an almost square Banach space with any Banach space is almost square, and the injective symmetric tensor product of an octahedral Banach space is octahedral.
The main aim of this note is to introduce the notion of an almost anti-periodic function in Banach space. We prove some characterizations for this class of functions, investigating also its relationship with the classes of anti-periodic functions and almost periodic functions in Banach spaces.
By introducing the concepts of order almost Dunford-Pettis and almost weakly limited operators in Banach lattices, we give some properties of them related to some well known classes of operators, such as, order weakly compact, order Dunford-Pettis, weak and almost Dunford-Pettis and weakly limited operators. Then, we characterize Banach lattices E and F on which each operator from E into F that is order almost Dunford-Pettis and weak almost Dunford-Pettis is an almost weakly limited operator.
The aim of this paper is to show that almost greedy bases induce tighter embeddings in superreflexive Banach spaces than in general Banach spaces. More specifically, we show that an almost greedy basis in a superreflexive Banach space $\mathbb{X}$ induces embeddings that allow squeezing $\mathbb{X}$ between two superreflexive Lorentz sequence spaces that are close to each other in the sense that they have the same fundamental function.
We show that finite dimensional Banach spaces fail to be uniformly non locally almost square. Moreover, we construct an equivalent almost square bidual norm on $\ell_\infty.$ As a consequence we get that every dual Banach space containing $c_0$ has an equivalent almost square dual norm. Finally we characterize separable real almost square spaces in terms of their position in their fourth duals.
A Banach space is locally almost square if, for every $y$ in its unit sphere, there exists a sequence $(x_n)$ in its unit sphere such that $\lim\|y\pm x_n\|=1$. A Banach space is weakly almost square if, in addition, we require the sequence $(x_n)$ to be weakly null. It is known that these two properties are distinct, so we aim to investigate if local almost squareness implies a weaker version of the latter property by replacing the sequence with a net. In order to achieve this result, we restrict ourselves to Banach lattices and introduce a strengthening of local almost squareness by requirin…
We analyze the ergodic properties of power-bounded operators on a reflexive Banach space of the form "scalar plus compact-power", and show that they are almost periodic (all the orbits are conditionally compact). If such an operator is weakly mixing, then it is stable (its powers converge in the strong operator topology).Let X-ISP be the separable reflexive indecomposable Banach space constructed by Argyros and Motakis, in which every operator has an invariant subspace. We conclude that every power-bounded operator on a closed subspace of X-ISP is almost periodic.
Let WAP(A) be the space of all weakly almost periodic functionals on a Banach algebra A. The Banach algebra A for which the natural embedding of A into WAP(A)* is bounded below is called a WAP-algebra. We show that the second dual of a Banach algebra A is a WAP-algebra, under each Arens products, if and only if A** is a dual Banach algebra. This is equivalent to the Arens regularity of A. For a locally compact foundation semigroup S, we show that the absolutely continuous semigroup measure algebra M_a(S) is a WAP-algebra if and only if the measure algebra M_b(S) is so.
We single out and study a natural class of Banach spaces -- almost square Banach spaces. In an almost square space we can find, given a finite set $x_1,x_2,\ldots,x_N$ in the unit sphere, a unit vector $y$ such that $\|x_i-y\|$ is almost one. These spaces have duals that are octahedral and finite convex combinations of slices of the unit ball of an almost square space have diameter 2. We provide several examples and characterizations of almost square spaces. We prove that non-reflexive spaces which are M-ideals in their biduals are almost square. We show that every separable space containing a…
A continuous operator $T$ between two Banach lattices $E$ and $F$ is called almost order-weakly compact, whenever for each almost order bounded subset $A$ of $E$, $T(A)$ is a relatively weakly compact subset of $F$. In Theorem 4, we show that the positive operator $T$ from $E$ into Dedekind complete $F$ is almost order-weakly compact if and only if $T(x_n) \xrightarrow{\|.\|}0$ in $F$ for each disjoint almost order bounded sequence $\{x_n\}$ in $E$. In this manuscript, we study some properties of this class of operators and its relationships with others known operators.
The first part of this article is a brief survey of the properties of so-called almost interior points in ordered Banach spaces. Those vectors can be seen as a generalization of ``functions which are strictly positive almost everywhere'' on $L^p$-spaces and of ``quasi-interior points'' in Banach lattices. In the second part we study the long--term behaviour of strongly positive operator semigroups on ordered Banach spaces; these are semigroups which, in a sense, map every non-zero positive vector to an almost interior point. Using the Jacobs--de Leeuw--Glicksberg decomposition together with th…
We prove some results on weakly almost square Banach spaces and their relatives. On the one hand, we discuss weak almost squareness in the setting of Banach function spaces. More precisely, let $(Ω,Σ)$ be a measurable space, let $E$ be a Banach lattice and let $ν:Σ\to E^+$ be a non-atomic countably additive measure having relatively norm compact range. Then the space $L_1(ν)$ is weakly almost square. This result applies to some abstract Cesàro function spaces. Similar arguments show that the Lebesgue-Bochner space $L_1(μ,Y)$ is weakly almost square for any Banach space~$Y$ and for any non-atom…
It is known that a Banach space contains an isomorphic copy of $c_0$ if, and only if, it can be equivalently renormed to be almost square. We introduce and study transfinite versions of almost square Banach spaces with the purpose to relate them to the containment of isomorphic copies of $c_0(κ)$, where $κ$ is some uncountable cardinal. We also provide several examples and stability results of the above properties by taking direct sums, tensor products and ultraproducts. By connecting the above properties with transfinite analogues of the strong diameter two property and octahedral norms, we o…
We introduce and study the notion of space of almost universal complemented disposition (a.u.c.d.) as a generalization of Kadec space. We show that every Banach space with separable dual is isometrically contained as a $1$-complemented subspace of a separable a.u.c.d. space and that all a.u.c.d. spaces with $1$-FDD are isometric and contain isometric $1$-complemented copies of every separable Banach space with $1$-FDD. We then study spaces of universal complemented disposition (u.c.d.) and provide different constructions for such spaces. We also consider spaces of u.c.d. with respect to separa…
We give a positive answer to the question of K. Bouras [`Almost Dunford-Pettis sets in Banach lattices', \textit{Rend. Circ. Mat. Palermo (2)} \textbf{ 62} (2013), 227--236] concerning weak compactness of almost Dunford-Pettis sets in Banach lattices. That is, every almost Dunford-Pettis set in a Banach lattice $E$ is relatively weakly compact if, and only if, $E$ is a $KB$-space.
In [3] it was proved that almost-greedy and semi-greedy bases are equivalent in the context of Banach spaces with finite cotype. In this paper we show this equivalence for general Banach spaces.
It is known that for a conditional quasi-greedy basis $\mathcal{B}$ in a Banach space $\mathbb{X}$, the associated sequence $(k_{m}[\mathcal{B}])_{m=1}^{\infty}$ of its conditionality constants verifies the estimate $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)$ and that if the reverse inequality $\log m =\mathcal{O}(k_m[\mathcal{B}])$ holds then $\mathbb{X}$ is non-superreflexive. Indeed, it is known that a quasi-greedy basis in a superreflexive quasi-Banach space fulfils the estimate $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)^{1-ε}$ for some $ε>0$. However, in the existing literature one finds very f…
In this paper, we study some geometric properties in Banach lattices and the class of almost limited completely continuous operators. For example, we study Banach lattices with the property (d) and we give a new characterization of this property concerning the solid hull of almost limited sets.
The main purpose of this paper is to introduce the notion of an asymptotically almost periodic ultradistribution and asymptotically almost automorphic ultradistribution with values in a Banach space, as well as to further analyze the classes of asymptotically almost periodic and asymptotically almost automorphic distributions with values in a Banach space. We provide some applications of the introduced concepts in the analysis of systems of ordinary differential equations.
A recent result of T.~Abrahamsen, P.~Hájek and S.~Troyanski states that a separable Banach space is almost square if and only if there exists $h\in S_{X^{****}}$ such that $\|x+h\|=\max\{\|x\|,1\}$ for all $x\in X$. The proof passes through a sequential version of being almost square which we call being \textit{sequentially almost square}. In this article we study these conditions in the nonseparable setting. On one hand, we show that a Banach space $X$ contains a copy of $c_0$ if and only if there exists an equivalent renorming $| \cdot |$ on $X$ for which there exists $h\in S_{X^{****}}$ suc…
We introduce the notion of a \textit{weight-almost greedy} basis and show that a basis for a real Banach space is $w$-almost greedy if and only if it is both quasi-greedy and $w$-democratic. We also introduce the notion of \textit{weight-semi-greedy} basis and show that a $w$-almost greedy basis is $w$-semi-greedy and that the converse holds if the Banach space has finite cotype.
In contrast with the separable case, we prove that the existence of almost $L$-orthogonal vectors in a nonseparable Banach space $X$ (octahedrality) does not imply the existence of nonzero vectors in $X^{**}$ being $L$-orthogonal to $X$, which shows that the answer to an environment question in [9] is negative. Furthermore, we prove that the abundance of almost $L$-orthogonal vectors in a Banach space $X$ (almost Daugavet property) whose density character is $ω_1$ implies the abundance of nonzero vectors in $X^{**}$ being $L$-orthogonal to $X$. In fact, we get that a Banach space $X$ whose den…
The main aim of this paper is to consider the classes of quasi-asymptotically almost periodic functions and Stepanov quasi-asymptotically almost periodic functions in Banach spaces. These classes extend the well known classes of asymptotically almost periodic functions, Stepanov asymptotically almost periodic functions and S-asymptotically $ω$-periodic functions with values in Banach spaces. We investigate the invariance of introduced properties under the action of finite and inifinite convolution products, providing also an illustrative application to abstract nonautonomous differential equat…
If $T$ is a bounded linear operator acting on an infinite-dimensional Banach space $X$, we say that a closed subspace $Y$ of $X$ of both infinite dimension and codimension is an almost-invariant halfspace (AIHS) under $T$ whenever $TY\subseteq Y+E$ for some finite-dimensional subspace $E$, or, equivalently, $(T+F)Y\subseteq Y$ for some finite-rank perturbation $F:X\to X$. We discuss the existence of AIHS's for various restrictions on $E$ and $F$ when $X$ is a complex Banach space. We also extend some of these and other results in the literature to the setting where $X$ is a real Banach space i…
We consider uncountable almost disjoint families of subsets of $\mathbb N$, the Johnson-Lindenstrauss Banach spaces $(\mathcal X_{\mathcal A}, \|\ \|_\infty)$ induced by them, and their natural equivalent renormings $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$. We introduce a partial order $\mathbb P_{\mathcal A}$ and characterize some geometric properties of the spheres of $(\mathcal X_{\mathcal A}, \|\ \|_{\infty})$ and of $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$ in terms of combinatorial properties of $\mathbb P_{\mathcal A}$. Exploiting the extreme behavior of some known and some…
We analyze a convex stochastic optimization problem where the state is assumed to belong to the Bochner space of essentially bounded random variables with images in a reflexive and separable Banach space. For this problem, we obtain optimality conditions that are, with an appropriate model, necessary and sufficient. Additionally, the Lagrange multipliers associated with optimality conditions are integrable vector-valued functions and not only measures. A model problem is given demonstrating the application to PDE-constrained optimization under uncertainty with an outlook for further applicatio…
Let $U_{FNA}$ be the class of all non-archimedean finite-dimensional Banach spaces. A non-archimedean Gurarii Banach space $G$ over a non-archimedean valued field $K$ is constructed, i.e. a non-archimedean Banach space $G$ of countable type which is of almost universal disposition for the class $U_{FNA}$. This means: for every isometry $g : X \to Y$ , where $Y \in U_{FNA}$ and $X$ is a subspace of $G$, and every $\varepsilon \in (0,1)$ there exists an $\varepsilon$-isometry $f : Y \to G$ such that $f(g(x)) = x$ for all $x \in X$. We show that all non-archimedean Banach spaces of almost univers…
The objective of this paper is to introduce the notion of generalized almost statistical (briefly, GAS) convergence of bounded real sequences, which generalizes the notion of almost convergence as well as statistical convergence of bounded real sequences. As a special kind of Banach limit functional, we also introduce the concept of Banach statistical limit functional and the notion of GAS convergence mainly depends on the existence of Banach statistical limit functional. We prove the existence of Banach statistical limit functional. Then we have shown the existence of a GAS convergent sequenc…
We consider a certain type of geometric properties of Banach spaces, which includes for instance octahedrality, almost squareness, lushness and the Daugavet property. For this type of properties, we obtain a general reduction theorem, which, roughly speaking, states the following: if the property in question is stable under certain finite absolute sums (for example finite $\ell^p$-sums), then it is also stable under the formation of corresponding Köthe-Bochner spaces (for example $L^p$-Bochner spaces). From this general theorem, we obtain as corollaries a number of new results as well as some …
We prove that in any Banach space the set of windows in which a rectifiable curve resembles two or more straight line segments is quantitatively small with constants that are independent of the curve, the dimension of the space, and the choice of norm. Together with Part I, this completes the proof of the necessary half of the Analyst's Traveling Salesman theorem with sharp exponent in uniformly convex spaces.
