Class Of Banach Spaces Research Articles

A Short Proof of the Uniform Smoothness of Certain Lebesgue Spaces

This note gives a short, direct proof of the uniform smoothness of Lp spaces with . Current proofs in the literature use the uniform convexity of the Lp spaces and the duality between convexity and smoothness. As a corollary, using duality, we obtain the uniform convexity of the Lp spaces with , which is the notoriously “difficult” range, without recourse to Clarkson’s second inequality. The method of proof also gives the power-type of the modulus of smoothness of Lp spaces with ; this is known and was stated by J. Lindenstrauss and L. Tzafriri in Classical Banach Spaces II in Chapter 1, Section e. The proof was postponed to Vol. III, but this was never written.

Local Uniform Kadec-Klee Property (LUKK) and Modulus of (LUKK)

A new geometry property and two new moduli are introduced in Banach space. First, the concept of local uniform Kadec-Klee property (LUKK) is introduced and the implication relationships between LUKK and local near uniform convexity LNUC, uniformly Kadec-Klee (UKK), (H) are investigated in Banach space. Furthermore, the modulus PXLε of (LUKK) and the modulus ΔXLε of LNUC are introduced and the relationship of size between PXLε and ΔXLε is also investigated in Banach space. Finally, several formulas for PXLε are calculated in classical Banach space lp.

Ulam Stability of a Functional Equation in Various Normed Spaces

In this note, we study the Ulam stability of a general functional equation in four variables. Since its particular case is a known equation characterizing the so-called bi-quadratic mappings (i.e., mappings which are quadratic in each of their both arguments), we get in consequence its stability, too. We deal with the stability of the considered functional equations not only in classical Banach spaces, but also in 2-Banach and complete non-Archimedean normed spaces. To obtain our outcomes, the direct method is applied.

On a rho -Orthogonally Additive Mappings

We show that a real normed linear space endowed with the rho -orthogonality relation, in general need not be an orthogonality space in the sense of Rätz. However, we prove that rho -orthogonally additive mappings defined on some classical Banach spaces have to be additive. Moreover, additivity (and approximate additivity) under the condition of an approximate orthogonality is considered.

V-Proximal Trustworthy Banach Spaces

In a recent work (2016), the first author proved the fuzzy sum rule for the V-proximal subdifferential under some natural assumptions on an equivalent norm of the Banach spaces. In the present paper, we are going to prove that the class of Banach spaces satisfying the fuzzy sum rule is very large and contains all Lp spaces 1<p<∞ as well as the sequence spaces lp1<p<∞, the Sobolev spaces Wp,n1<p<∞, and the Schatten trace ideals Cp1<p<∞.

A last theorem of Kalton and finiteness of Connes' integral

We connect finiteness of the noncommutative integral in Alain Connes' noncommutative geometry with the study of tensor multipliers from classical Banach space theory. For the Lorentz function spaceΛ1(Rd)={f∈L0(Rd):∫0∞μ(s,f)(1+log+⁡(s−1))ds<∞} where μ(s,f), s>0, denotes the decreasing rearrangement of f, and log+ denotes the positive part of log on (0,∞), we prove using tensor multipliers the formulaφ((1−ΔRd)−d/4Mf(1−ΔRd)−d/4)=VolSd−1d(2π)d∫Rdf(x)dx,f∈Λ1(Rd). Here −ΔRd is the selfadjoint extension of minus the Laplacian on Rd, Mf denotes the operation of pointwise multiplication, the operator (1−ΔRd)−d/4Mf(1−ΔRd)−d/4 has a bounded extension which is a compact operator from the Hilbert space L2(Rd) to itself, and φ is any continuous normalised trace on the ideal of compact operators on L2(Rd) with series of singular values at most logarithmically diverge. The formula fails given only f∈L1(Rd), and previously had been shown by different methods for the smaller set of functions f∈L2(Rd) that have compact support.We prove a similar formula for the Laplace-Beltrami operator on a compact Riemannian manifold without boundary.We discuss how the integral formula incorporates a last theorem of Nigel Kalton. We also extend to the case p=2 a classical result of Cwikel on weak estimates p>2 of operators of the form Mfg(−i∇), f∈Lp(Rd), g∈Lp,∞(Rd) where ∇ is the gradient operator.

Sequentially Right Banach spaces of order $p$

We introduce and study two new classes of Banach spaces, the so-called sequentially Right Banach spaces of order $p$, and those defined by the dual property, the sequentially Right$^*$ Banach spaces of order $p$ for $1\leq p\leq\infty$. These classes of Banach spaces are characterized by the notions of $L_p$-limited sets in the corresponding dual space and $R^*_p$ subsets of the involved Banach space, respectively. In particular, we investigate whether the injective tensor product of a Banach space $X$ and a reflexive Banach space $Y$ has the sequentially Right property of order $p$ when $X$ enjoys this property.

Strategically reproducible bases and the factorization property

We introduce the concept of strategically reproducible bases in Banach spaces and show that operators which have large diagonal with respect to strategically reproducible bases are factors of the identity. We give several examples of classical Banach spaces in which the Haar system is strategically reproducible: multi-parameter Lebesgue spaces, mixed-norm Hardy spaces and most significantly the space $L^1$. Moreover, we show the strategical reproducibility is inherited by unconditional sums.

Twisted sums of c0 and C(K)-spaces: A solution to the CCKY problem

We consider the class of Banach spaces Y for which c0 admits a nontrivial twisted sum with Y. We present a characterization of such spaces Y in terms of properties of the weak⁎ topology on Y⁎. We prove that under the continuum hypothesis c0 has a nontrivial twisted sum with every space of the form Y=C(K), where K is compact and not metrizable. This gives a consistent affirmative solution to a problem posed by Cabello, Castillo, Kalton and Yost.

Stability results of properties related to the Bishop-Phelps-Bollobás property for operators

We prove that the class of Banach spaces $Y$ such that the pair $(\ell_1, Y)$ has the Bishop-Phelps-Bollob\'as property for operators is stable under finite products when the norm of the product is given by an absolute norm. We also provide examples showing that previous stability results obtained for that property are optimal.

COMPUTATION OF ZEROS OF MONOTONE TYPE MAPPINGS: ON CHIDUME’S OPEN PROBLEM

For $p\geq 2$, let $E$ be a 2-uniformly smooth and $p$-uniformly convex real Banach space and let $A:E\rightarrow E^{\ast }$ be a Lipschitz and strongly monotone mapping such that $A^{-1}(0)\neq \emptyset$. For given $x_{1}\in E$, let $\{x_{n}\}$ be generated by the algorithm $x_{n+1}=J^{-1}(Jx_{n}-\unicode[STIX]{x1D706}Ax_{n})$, $n\geq 1$, where $J$ is the normalized duality mapping from $E$ into $E^{\ast }$ and $\unicode[STIX]{x1D706}$ is a positive real number in $(0,1)$ satisfying suitable conditions. Then it is proved that $\{x_{n}\}$ converges strongly to the unique point $x^{\ast }\in A^{-1}(0)$. Furthermore, our theorems provide an affirmative answer to the Chidume et al. open problem [‘Krasnoselskii-type algorithm for zeros of strongly monotone Lipschitz maps in classical Banach spaces’, SpringerPlus4 (2015), 297]. Finally, applications to convex minimization problems are given.

Straightening warped cones

We provide the converses to two results of Roe [Warped cones and property A, Geom. Topol. 9 (2005) 163–178, doi:10.2140/9t.2005.9.163]: first, the warped cone associated to a free action of an a-T-menable group admits a fibered coarse embedding into a Hilbert space, and second, a free action yielding a warped cone with property A must be amenable. We construct examples showing that in both cases the freeness assumption is necessary. The first equivalence is obtained also for other classes of Banach spaces, in particular for [Formula: see text]-spaces.

Non-superreflexivity of Garling sequence spaces and applications to the existence of special types of conditional bases

We settle in the negative the problem of the superreflexivity of Garling sequence spaces by showing that they contain a complemented subspace isomorphic to a non-superreflexive mixed-norm sequence space. As a by-product, we give applications to the study

A nonlinear representation of dual spaces and its applications

We give a nonlinear representation of the duals for a class of Banach spaces. This leads to classroom-friendly proofs of the classical representation theorems $$H' = H {\rm and} (L^{p})' = L^{q}$$. Our proofs extend to a family of Orlicz spaces, and yield as an unexpected byproduct a version of the Helly–Hahn–Banach theorem.

Light groups of isomorphisms of Banach spaces and invariant LUR renormings

Megrelishvili defines \emph{light groups} of isomorphisms of a Banach space as the groups on which the Weak and Strong Operator Topologies coincide, and proves that every bounded group of isomorphisms of Banach spaces with the Point of Continuity Property (PCP) is light. We investigate this concept for isomorphism groups $G$ of classical Banach spaces $X$ without the PCP, specially isometry groups, and relate it to the existence of $G$-invariant LUR or strictly convex renormings of $X$.

Coorbits for projective representations with an application to Bergman spaces

Representation theory of locally compact topological groups is a powerful tool to analyze Banach spaces of functions and distributions. It provides a unified framework for constructing function spaces and to study several generalizations of the wavelet transform. Recently representation theory has been used to provide atomic decompositions for a large collection of classical Banach spaces. But in some natural situations, including Bergman spaces on bounded domains, representations are too restrictive. The proper tools are projective representations. In this paper we extend known techniques from representation theory to also include projective representations. This leads naturally to twisted convolution on groups avoiding the usual central extension of the group. As our main application we obtain atomic decompositions of Bergman spaces on the unit ball through the holomorphic discrete series for the group of isometries of the ball.

Nonpositive curvature is not coarsely universal

We prove that not every metric space embeds coarsely into an Alexandrov space of nonpositive curvature. This answers a question of Gromov (1993) and is in contrast to the fact that any metric space embeds coarsely into an Alexandrov space of nonnegative curvature, as shown by Andoni, Naor and Neiman (2015). We establish this statement by proving that a metric space which is $q$-barycentric for some $q\in [1,\infty)$ has metric cotype $q$ with sharp scaling parameter. Our proof utilizes nonlinear (metric space-valued) martingale inequalities and yields sharp bounds even for some classical Banach spaces. This allows us to evaluate the bi-Lipschitz distortion of the $\ell_\infty$ grid $[m]_\infty^n=(\{1,\ldots,m\}^n,\|\cdot\|_\infty)$ into $\ell_q$ for all $q\in (2,\infty)$, from which we deduce the following discrete converse to the fact that $\ell_\infty^n$ embeds with distortion $O(1)$ into $\ell_q$ for $q=O(\log n)$. A rigidity theorem of Ribe (1976) implies that for every $n\in \mathbb{N}$ there exists $m\in \mathbb{N}$ such that if $[m]_\infty^n$ embeds into $\ell_q$ with distortion $O(1)$, then $q$ is necessarily at least a universal constant multiple of $\log n$. Ribe's theorem does not give an explicit upper bound on this $m$, but by the work of Bourgain (1987) it suffices to take $m=n$, and this was the previously best-known estimate for $m$. We show that the above discretization statement actually holds when $m$ is a universal constant.

Some classes of Banach spaces and complemented subspaces of operators

‎The concept of $p$-$L$-limited sets and Banach spaces with the $p$-$L$-limited property ($p\in [1, \infty)$) are studied‎. ‎Some characterizations of limited $p$-convergent operators are obtained‎. ‎The complementability of some spaces of operators in the space of limited $p$-convergent operators is also investigated‎.

On a second numerical index for Banach spaces

AbstractWe introduce a second numerical index for real Banach spaces with non-trivial Lie algebra, as the best constant of equivalence between the numerical radius and the quotient of the operator norm modulo the Lie algebra. We present a number of examples and results concerning absolute sums, duality, vector-valued function spaces…which show that, in many cases, the behaviour of this second numerical index differs from the one of the classical numerical index. As main results, we prove that Hilbert spaces have second numerical index one and that they are the only spaces with this property among the class of Banach spaces with one-unconditional basis and non-trivial Lie algebra. Besides, an application to the Bishop-Phelps-Bollobás property for the numerical radius is given.

The Bishop-Phelps-Bollobás modulus for functionals on classical Banach spaces

‎In this manuscript, we compute the Bishop-Phelps-Bollobas modulus for functionals in classical Banach spaces‎, ‎such as Hilbert spaces‎, ‎spaces of continuous functions $c_0$ and $\ell_1$‎.