Class Of Banach Spaces Research Articles

Stability criteria and calculus rules via conic contingent coderivatives in Banach spaces

This paper addresses the study of novel constructions of variational analysis and generalized differentiation that are appropriate for characterizing robust stability properties of constrained set-valued mappings/multifunctions between Banach spaces important in optimization theory and its applications. Our tools of generalized differentiation revolve around the newly introduced concept of ε-regular normal cone to sets and associated coderivative notions for set-valued mappings. Based on these constructions, we establish several characterizations of the central stability notion known as the relative Lipschitz-like property of set-valued mappings in infinite dimensions. Applying a new version of the constrained extremal principle of variational analysis, we develop comprehensive sum and chain rules for our major constructions of conic contingent coderivatives for multifunctions between appropriate classes of Banach spaces.

Linear versus nonlinear forms of partial unconditionality of bases

The main results in this paper contribute to bringing to the fore novel underlying connections between the contemporary concepts and methods springing from greedy approximation theory with the well-established techniques of classical Banach spaces. We do that by showing that bounded-oscillation unconditional bases, introduced by Dilworth et al. in 2009 in the setting of their search for extraction principles of subsequences verifying partial forms of unconditionality, are the same as truncation quasi-greedy bases, a new breed of bases that appear naturally in the study of the performance of the thresholding greedy algorithm in Banach spaces. We use this identification to provide examples of bases that exhibit that bounded-oscillation unconditionality is a stronger condition than Elton's near unconditionality. We also take advantage of our arguments to provide examples that allow us to tell apart certain types of bases that verify either debilitated unconditionality conditions or weaker forms of quasi-greediness in the context of abstract approximation theory.

Banach spaces of [formula omitted]-convergent sequences

We study the space c0,I(X) of all bounded sequences (xn) in a Banach space X that I-converges to 0, endowed with the sup norm, where I is an ideal of subsets of N. Our results contribute to the development of a structural theory for these spaces. We show that two such spaces, c0,I and c0,J, are isometric exactly when the ideals I and J are isomorphic. Additionally, we analyze the connection of the well-known Katětov pre-order ≤K on ideals with some properties of the space c0,I. For instance, we show that I≤KJ exactly when there is a (not necessarily onto) Banach lattice isometry from c0,I to c0,J, satisfying some additional conditions.We present some lattice-theoretic properties of c0,I, particularly demonstrating that every closed ideal of ℓ∞ is equal to c0,I for some ideal I on N. We also show that certain classical Banach spaces are isometric to c0,I for some ideal I, such as the spaces ℓ∞(c0) and c0(ℓ∞). Finally, we provide several examples of ideals for which c0,I is not a Grothendieck space.

Żuk’s criterion for Banach spaces and random groups

Abstract We prove a Banach version of Żuk’s criterion for groups acting on partite (i.e., colorable) simplicial complexes. Using this new criterion, we derive a new fixed point theorem for random groups in the Gromov density model with respect to several classes of Banach spaces ( $L^p$ spaces, Hilbertian spaces, uniformly curved spaces). In particular, we show that for every p, a group in the Gromov density model has asymptotically almost surely property $(F L^p)$ and give a sharp lower bound for the growth of the conformal dimension of the boundary of such group as a function of the parameters of the density model.

Estimates for evolutionary partial differential equations in classical function spaces

Abstract We establish new local and global estimates for evolutionary partial differential equations in classical Banach and quasi-Banach spaces that appear most frequently in the theory of partial differential equations. More specifically, we obtain optimal (local in time) estimates for the solution to the Cauchy problem for variable-coefficient evolutionary partial differential equations. The estimates are achieved by introducing the notions of Schrödinger and general oscillatory integral operators with inhomogeneous phase functions and prove sharp local and global regularity results for these in Besov–Lipschitz and Triebel–Lizorkin spaces.

A generalized Halpern-type forward-backward splitting algorithm for solving variational inclusion problems

<abstract><p>In this paper, we investigate the problem of finding a zero of sum of two accretive operators in the setting of uniformly convex and $ q $-uniformly smooth real Banach spaces ($ q > 1 $). We incorporate the inertial and relaxation parameters in a Halpern-type forward-backward splitting algorithm to accelerate the convergence of its sequence to a zero of sum of two accretive operators. Furthermore, we prove strong convergence of the sequence generated by our proposed iterative algorithm. Finally, we provide a numerical example in the setting of the classical Banach space $ l_4(\mathbb{R}) $ to study the effect of the relaxation and inertial parameters in our proposed algorithm.</p></abstract>

Factorisation in stopping-time Banach spaces: Identifying unique maximal ideals

Stopping-time Banach spaces is a collective term for the class of spaces of eventually null integrable processes that are defined in terms of the behaviour of the stopping times with respect to some fixed filtration. From the point of view of Banach space theory, these spaces in many regards resemble the classical spaces such as L1 or C(Δ), but unlike these, they do have unconditional bases. In the present paper, we study the canonical bases in the stopping-time spaces in relation to factorising the identity operator thereon. Since we work exclusively with the dyadic-tree filtration, this setup enables us to work with tree-indexed bases rather than directly with stochastic processes. En route to the factorisation results, we develop general criteria that allow one to deduce the uniqueness of the maximal ideal in the algebra of operators on a Banach space. These criteria are applicable to many classical Banach spaces such as (mixed-norm) Lp-spaces, BMO, SL∞, and others.

Uniqueness of Hahn–Banach extensions and some of its variants

In this paper, we analyze the various strengthening and weakening of the uniqueness of the Hahn–Banach extension. In addition, we consider the case in which Y is an ideal of X. In this context, we study the property (U)/(SU)/(HB) and property (wU)/(k-U) for a subspace Y of a Banach space X. We obtain various new characterizations of these properties. We study different kinds of stabilities resulting from these properties in the tensor product spaces, spaces of Bochner integrable functions, and the higher duals of Banach spaces. We discuss various examples in the classical Banach spaces, where the aforementioned properties are satisfied and where they fail. It is established that a hyperplane in \(c_0\) has property (HB) if and only if it is an M-summand. It is observed that a finite-dimensional subspace Y has property (k-U) in \(c_0\), in addition to that if Y is an ideal, then \(Y^*\) is a k-strictly convex subspace of \(\ell _1\) for some natural k.

The embedding property of the scaling limit of modulation spaces

Modulation spaces Mp,qs were introduced by Feichtinger [3] in 1983. Later, considering the scaling property of the modulation spaces, Sugimoto and Wang [11] defined the scaling limit of the modulation spaces, which contains both the modulation spaces and Bényi and Oh's modulation spaces [1], and these spaces also have some applications in nonlinear Schrödinger equations. So, it is important to consider the relationship between these new spaces and some classical Banach spaces such as Lp spaces, Fourier Lp spaces and Besov-Triebel-Sobolev spaces. We study the embedding properties of the scaling limit of the modulation spaces, including the homogeneous case and nonhomogeneous case.

Norm-Attaining Tensors and Nuclear Operators

Given two Banach spaces X and Y, we introduce and study a concept of norm-attainment in the space of nuclear operators \(\mathcal N(X,Y)\) and in the projective tensor product space \(X\widehat{\otimes }_\pi Y\). We exhibit positive and negative examples where both previous norm-attainment hold. We also study the problem of whether the class of elements which attain their norms in \(\mathcal N(X,Y)\) and in \(X\widehat{\otimes }_\pi Y\) is dense or not. We prove that, for both concepts, the density of norm-attaining elements holds for a large class of Banach spaces X and Y which, in particular, covers all classical Banach spaces. Nevertheless, we present Banach spaces X and Y failing the approximation property in such a way that the class of elements in \(X\widehat{\otimes }_\pi Y\) which attain their projective norms is not dense. We also discuss some relations and applications of our work to the classical theory of norm-attaining operators throughout the paper.

Stochastic integration with respect to fractional processes in Banach spaces

In the article, integration of temporal functions in (possibly non-UMD) Banach spaces with respect to (possibly non-Gaussian) fractional processes from a finite sum of Wiener chaoses is treated. The family of fractional processes that is considered includes, for example, fractional Brownian motions of any Hurst parameter or, more generally, fractionally filtered generalized Hermite processes. The class of Banach spaces that is considered includes a large variety of the most commonly used function spaces such as the Lebesgue spaces, Sobolev spaces, or, more generally, the Besov and Lizorkin-Triebel spaces. In the article, a characterization of the domains of the Wiener integrals on both bounded and unbounded intervals is given for both scalar and cylindrical fractional processes. In general, the integrand takes values in the space of γ-radonifying operators from a certain homogeneous Sobolev-Slobodeckii space into the considered Banach space. Moreover, an equivalent characterization in terms of a pointwise kernel of the integrand is also given if the considered Banach space is isomorphic with a subspace of a cartesian product of mixed Lebesgue spaces. The results are subsequently applied to stochastic convolution for which both necessary and sufficient conditions for measurability and sufficient conditions for continuity are found. As an application, space-time continuity of the solution to a parabolic equation of order 2m with distributed noise of low time regularity is shown as well as measurability of the solution to the heat equation with Neumann boundary noise of higher regularity.

Spaceability on some classes of Banach spaces

In this paper, we study spaceability of subsets of generalized Orlicz and Lebesgue spaces associated to Banach function space. Also, we give some sufficient conditions for spaceability of subsets of a general Banach space which improves an important result on this topic. As an application, it is shown that the set of all bounded linear operators which are not positive semidefinite on a separable Hilbert space is spaceable.

Characterizations of Integral Type for Weighted Classes of Analytic Banach Function Spaces

The aim of this study is to give some new definitions of Banach spaces of holomorphic functions. Some holomorphic characterizations of integral type for some classes of Banach spaces of holomorphic functions are established in the unit disc U.

The space is primary for 1 < p < ∞

Abstract The classical Banach space $L_1(L_p)$ consists of measurable scalar functions f on the unit square for which $$ \begin{align*}\|f\| = \int_0^1\Big(\int_0^1 |f(x,y)|^p dy\Big)^{1/p}dx < \infty.\end{align*} $$ We show that $L_1(L_p) (1 < p < \infty )$ is primary, meaning that whenever $L_1(L_p) = E\oplus F$ , where E and F are closed subspaces of $L_1(L_p)$ , then either E or F is isomorphic to $L_1(L_p)$ . More generally, we show that $L_1(X)$ is primary for a large class of rearrangement-invariant Banach function spaces.

Bishop–Phelps cones given by an equation in Banach spaces

In this work, we study Bishop–Phelps cones (briefly, BP cones) given by an equation in Banach spaces. Due to the special form, these cones enjoy interesting properties. We show that nontrivial BP cones have given by an equation form a ‘large family’ in some sense in any Banach space and they can be used to characterize the strict convexity of the space. We obtain conditions for a convex cone to be included in or to be itself a BP cone given by an equation. Furthermore, we give an affirmative answer to the open question of whether these cones may possess a nonempty interior in infinite-dimensional spaces and introduce Lorentz cones in some classical Banach spaces of sequences as illustrations. We also obtain explicit representations for all BP cones given by an equation in some classical Banach spaces. Finally, we present some short applications in optimal control and approximation.

Lipschitz integral operators represented by vector measures

<p>In this paper we introduce the concept of Lipschitz Pietsch-p-integral <br />mappings, (1≤p≤∞), between a metric space and a Banach space. We represent these mappings by an integral with respect to a vector<br />measure defined on a suitable compact Hausdorff space, obtaining in this way a rich factorization theory through the classical Banach spaces C(K), L_p(μ,K) and L_∞(μ,K). Also we show that this type of operators fits in the theory of composition Banach Lipschitz operator ideals. For p=∞, we characterize the Lipschitz Pietsch-∞-integral mappings by a factorization schema through a weakly compact operator. Finally, the relationship between these mappings and some well known Lipschitz operators is given.</p>

Covariant functions of characters of compact subgroups

This paper presents a systematic study for abstract harmonic analysis on classical Banach spaces of covariant functions of characters of compact subgroups. Let G be a locally compact group and H be a compact subgroup of G. Suppose that xi :Hrightarrow mathbb {T} is a character, 1le p<infty and L_xi ^p(G,H) is the set of all covariant functions of xi in L^p(G). It is shown that L^p_xi (G,H) is isometrically isomorphic to a quotient space of L^p(G). It is also proven that L^q_xi (G,H) is isometrically isomorphic to the dual space L^p_xi (G,H)^*, where q is the conjugate exponent of p. The paper is concluded by some results for the case that G is compact.

Classifying the closed ideals of bounded operators on two families of non-separable classical Banach spaces

We classify the closed ideals of bounded operators acting on the Banach spaces (⨁n∈Nℓ2n)c0⊕c0(Γ) and (⨁n∈Nℓ2n)ℓ1⊕ℓ1(Γ) for every uncountable cardinal Γ.

A short proof that ℬ(L1) is not amenable

Non-amenability of ${\mathcal {B}}(E)$ has been surprisingly difficult to prove for the classical Banach spaces, but is now known for E = ℓp and E = Lp for all 1 ⩽ p &lt; ∞. However, the arguments are rather indirect: the proof for L1 goes via non-amenability of $\ell ^\infty ({\mathcal {K}}(\ell _1))$ and a transference principle developed by Daws and Runde (Studia Math., 2010).In this note, we provide a short proof that ${\mathcal {B}}(L_1)$ and some of its subalgebras are non-amenable, which completely bypasses all of this machinery. Our approach is based on classical properties of the ideal of representable operators on L1, and shows that ${\mathcal {B}}(L_1)$ is not even approximately amenable.

Hanson–Wright inequality in Banach spaces

We discuss two-sided bounds for moments and tails of quadratic forms in Gaussian random variables with values in Banach spaces. We state a natural conjecture and show that it holds up to additional logarithmic factors. Moreover in a certain class of Banach spaces (including $L_r$-spaces) these logarithmic factors may be eliminated. As a corollary we derive upper bounds for tails and moments of quadratic forms in subgaussian random variables, which extend the Hanson-Wright inequality.