Class Of Banach Spaces Research Articles

The Generalized Empirical Interpolation Method: Stability theory on Hilbert spaces with an application to the Stokes equation

The Generalized Empirical Interpolation Method (GEIM) is an extension first presented by Maday and Mula in [1] in 2013 of the classical empirical interpolation method (presented in 2004 by Barrault, Maday, Nguyen and Patera in [2]) where the evaluation at interpolating points is replaced by the more practical evaluation at interpolating continuous linear functionals on a class of Banach spaces. As outlined in [1], this allows to relax the continuity constraint in the target functions and expand both the application domain and the stability of the approach. In this paper, we present a thorough analysis of the concept of stability condition of the generalized interpolant (the Lebesgue constant) by relating it to an inf-sup problem in the case of Hilbert spaces. In the second part of the paper, it will be explained how GEIM can be employed to monitor in real time physical experiments by providing an online accurate approximation of the phenomenon that is computed by combining the acquisition of a minimal number, optimally placed, measurements from the processes with their mathematical models (parameter-dependent PDEs). This idea is illustrated through a parameter dependent Stokes problem in which it is shown that the pressure and velocity fields can e ciently be reconstructed with a relatively low-dimensional interpolation space.

Algebraic and invariance properties of the group of isometries

In this paper we investigate the algebraic structure of the isometry group of several classical Banach spaces, namely, symmetric sequence spaces, Cp(H) (1≤p<∞,p≠2), L(ℓp,ℓr), 1≤p,r<∞, and bounded operators L(H) endowed with Chan–Li–Tui unitarily invariant (c,p)-norms. We also identify the isometrically invariant subspaces for each of theses settings.

Structural transition between $L^{p}(G)$ and $L^{p}(G/H)$

Let $H$ be a compact subgroup of a locally compact group $G$. We consider the homogeneous space $G/H$ equipped with a strongly quasi-invariant Radon measure $\mu$. For $1\leq p\leq +\infty$, we introduce a norm decreasing linear map from $L^{p}(G)$ onto $L^{p}(G/H,\mu)$ and show that $L^{p}(G/H,\mu)$ may be identified with a quotient space of $L^{p}(G)$. Also, we prove that $L^{p}(G/H,\mu)$ is isometrically isomorphic to a closed subspace of $L^{p}(G)$. These help us study the structure of the classical Banach spaces constructed on a homogeneous space via those created on topological groups.

Best Constrained Approximation in Banach Spaces

In this article, for a real Banach space X and a closed subspace Y, we consider aspects of proximinality, its stronger variants and the notion of centrability, Chebyshev centers, for a class of subspaces that are relatively constrained in a Banach space, in the sense that for x ∉ Y, Y is constrained in span{x, Y}. We show that if X, under the canonical embedding has the strong--ball property in its bidual, then the same is true of Y. We also give applications of these results to proximinality in spaces of Bochner integrable functions. We consider a class of Banach spaces for which a formula due to Smith and Ward on relative Chebyshev centers and radius is valid. We show that any locally constrained subspace of the Grothendieck G-space is a G-space.

The finite-dimensional decomposition property in non-Archimedean Banach spaces

A non-Archimedean Banach space has the orthogonal finite-dimensional decomposition property (OFDDP) if it is the orthogonal direct sum of a sequence of finite-dimensional subspaces. This property has an influence in the non-Archimedean Grothendieck’s approximation theory, where an open problem is the following: Let E be a non-Archimedean Banach space of countable type with the OFDDP and let D be a closed subspace of E. Does D have the OFDDP? In this paper we give a negative answer to this question; we construct a Banach space of countable type with the OFDDP having a one-codimensional subspace without the OFDDP. Next we prove that, however, for certain classes of Banach spaces of countable type, the OFDDP is preserved by taking finite-codimensional subspaces.

Isomorphic Schauder decompositions in certain Banach spaces

Abstract We extend a theorem of Kato on similarity for sequences of projections in Hilbert spaces to the case of isomorphic Schauder decompositions in certain Banach spaces. To this end we use ℓψ-Hilbertian and ∞-Hilbertian Schauder decompositions instead of orthogonal Schauder decompositions, generalize the concept of an orthogonal Schauder decomposition to the case of Banach spaces and introduce the class of Banach spaces with Schauder-Orlicz decompositions. Furthermore, we generalize the notions of type, cotype, infratype and M-cotype of a Banach space and study the properties of unconditional Schauder decompositions in Banach spaces possessing certain geometric structure.

A class of subspaces of Morrey spaces and norm inequalities on Riesz potential operators

We introduce two classes of Banach spaces $$F(q,p,\alpha )$$ and $$T^{p,\;\alpha }$$ ( $$1\le q\le \alpha \le p\le \infty $$ ) which are subspaces of Herz spaces and Morrey spaces of functions and measures respectively. We study the Riesz potential operators in these spaces and obtain norm inequalities on these operators from which we deduce a necessary condition for a nonnegative Radon measure to have its Riesz potential in a given Lebesgue space.

Compact lines and the Sobczyk property

We show that Sobczyk's Theorem holds for a new class of Banach spaces, namely spaces of continuous functions on linearly ordered compacta.

Isomorphic Universality and the Number of Pairwise Nonisomorphic Models in the Class of Banach Spaces

We develop the framework ofnatural spacesto study isomorphic embeddings of Banach spaces. We then use it to show that a sufficient failure of the generalized continuum hypothesis implies that the universality number of Banach spaces of a given density under a certain kind of positive embedding (very positive embedding) is high. An example of a very positive embedding is a positive onto embedding betweenC(K)andCLfor 0-dimensionalKandLsuch that the following requirement holds for allh≠0andf≥0inC(K): if0≤Th≤Tf, then there are constantsa≠0andbwith0≤a·h+b≤fanda·h+b≠0.

Cantor–Bernstein theorems for Orlicz sequence spaces

For two Banach spaces $X$ and $Y$, we write $\dim _\ell(X)= \dim _\ell(Y)$ if $X$ embeds into&nbsp;$Y$ and vice versa; then we say that \emph{$X$ and $Y$ have the same linear dimension}. In this paper, we consider classes of Banach spaces with symmetric b

On p-compact sets in classical Banach spaces

On p-compact sets in classical Banach spaces

Proximal Normal Cone Analysis on Smooth Banach Spaces and Applications

Unlike the Frechet, Penot, Clarke, and other “order one” notions of normal cones, the notion of proximal normal cone describes variational behavior of “order two,” and most of results on proximal normal cone are established in the Hilbert space framework. A possible reason is that in a general Banach space (such as in the classical Banach spaces $l^p$ and $L^p$ with $1 \leq p < 2$), the proximal normal cone of a “smooth” subset $A$ at every point of $A$ may consist of the zero element only. Nevertheless, on the other hand, we show in this paper that for a fairly big class of smooth Banach spaces (including the classical Banach spaces $l^p$ and $L^p$ with $2 \leq p <\infty$), the consideration of proximal normal cones provides a lot of useful information, especially for leading to satisfactory treatment for constrained optimization problems. Some of our results are new even in the Hilbert space case.

Recurrent Linear Operators

We study the notion of recurrence and some of its variations for linear operators acting on Banach spaces. We characterize recurrence for several classes of linear operators such as weighted shifts, composition operators and multiplication operators on classical Banach spaces. We show that on separable complex Hilbert spaces the study of recurrent operators reduces, in many cases, to the study of unitary operators. Finally, we study the notion of product recurrence and state some relevant open questions.

Reflexivity is equivalent to the perturbed fixed point property for cascading nonexpansive maps in Banach lattices

Using a theorem of Domínguez Benavides and the Strong James’ Distortion Theorems, we prove that if a Banach space is a Banach lattice, or has an unconditional basis, or is a symmetrically normed ideal of operators on an infinite-dimensional Hilbert space, then it is reflexive if and only if it has an equivalent norm that has the fixed point property for cascading nonexpansive mappings. This new class of mappings strictly includes nonexpansive mappings.Also, using a theorem of Mil’man and Mil’man, we show that for the above classes of Banach spaces, reflexivity is equivalent to the fixed point property for affine cascading nonexpansive mappings.Further, we show that a cascading nonexpansive mapping on a closed, bounded, convex set in a Banach space always has an approximate fixed point sequence.

ALMOST ISOMETRIC IDEALS IN BANACH SPACES

AbstractA natural class of ideals, almost isometric ideals, of Banach spaces is defined and studied. The motivation for working with this class of subspaces is our observation that they inherit diameter 2 properties and the Daugavet property. Lindenstrauss spaces are known to be the class of Banach spaces that are ideals in every superspace; we show that being an almost isometric ideal in every superspace characterizes the class of Gurariy spaces.

On positive cubature rules on the simplex and isometric embeddings

Positive cubature rules of degree 4 4 and 5 5 on the d d -dimensional simplex are constructed for a range of dimensions d d and used to construct cubature rules of index 8 8 or degree 9 9 on the unit sphere. The latter ones lead to explicit isometric embedding among the classical Banach spaces. Among other things, our results include several explicit representations of ( x 1 2 + ⋯ + x d 2 ) t (x_1^2+ \cdots + x_d^2)^t in terms of linear forms of degree 2 t 2t with rational coefficients for t = 4 t = 4 and 5 5 .

S. Bernstein’s idea for bounding the gradient of solutions to the quasi-linear Dirichlet problem

S. Bernstein’s idea is outlined to estimate \({\| u_{x} \|_{0,0,\Omega}}\) of a solution u(x) to the quasi-linear elliptic differential equation, especially in the case of n > 2. Up to about 1956, investigations of nonlinear problems have been dealing with the case of n = 2. A large number of methods were developed, but none of them were applicable for n > 2. The maximum-minimum principle had been a powerful tool to find bounds, however in the case of n > 2 this tool is not available for the gradient of the solution. In 1956, Cordes [Math. Ann. 130 (1956), 278–312] gave estimates for the case n > 2. Later, Ladyzhenskaya and Ural’tseva [Linear and Quasilinear Elliptic Equations. Academic Press, New York, 1968] established estimates in the Sobolev space \({W^{2}_{2} (\Omega)}\) . In their line of reasoning they used the idea of S. Bernstein and investigated the transformed function υ(x) with u(x) = ϕ(υ(x)). In this paper, we give a more simpler proof of those results in the classical Banach space C2,α(Ω).

Sharp corner points and isometric extension problem in Banach spaces

In this article, we begin using some geometric methods to study the isometric extension problem in general real Banach spaces. For any Banach space Y, we define a collection of “sharp corner points” of the unit ball B1(Y∗), which is empty if Y is strictly convex and dimY≥2. Then we prove that any surjective isometry between two unit spheres of Banach spaces X and Y has a linear isometric extension on the whole space if Y is a Gâteaux differentiability space (in particular, separable spaces or reflexive spaces) and the intersection of “sharp corner points” and weak∗-exposed points of B(Y∗) is weak∗-dense in the latter. Moreover, we study the “sharp corner points” in many classical Banach spaces and solve isometric extension problem affirmatively in the case that Y is (ℓ1),c0(Γ),c(Γ),ℓ∞(Γ) or some C(Ω).

Stability of vector measures and twisted sums of Banach spaces

A Banach space X is said to have the SVM (stability of vector measures) property if there exists a constant v<∞ such that for any algebra of sets F, and any function ν:F→X satisfying‖ν(A∪B)−ν(A)−ν(B)‖⩽1for disjoint A,B∈F, there is a vector measure μ:F→X with ‖ν(A)−μ(A)‖⩽v for all A∈F. If this condition is valid when restricted to set algebras F of cardinality less than some fixed cardinal number κ, then we say that X has the κ-SVM property. The least cardinal κ for which X does not have the κ-SVM property (if it exists) is called the SVM character of X. We apply the machinery of twisted sums and quasi-linear maps to characterise these properties and to determine SVM characters for many classical Banach spaces. We also discuss connections between the κ-SVM property, κ-injectivity and the ‘three-space’ problem.

Common fixed point of single valued and multivalued mappings

In this paper, we prove the existence of common fixed point of commuting and weakly commuting pair of single valued and multivalued mapping in some classes of Banach spaces. Our results extend previously known ones, we cites Theorem 2, Theorem 3 and Theorem 4 by Itoh and Takahashi (J Math Anal Appl 59(3):514–521, 1977), Theorem 4.2 by Kaewcharoen (Nonlinear Anal Hybrid Syst 4(3):389–394, 2010), Theorem 4.2 by Dhompongsa et al. (Nonlinear Anal 64(5):958–970, 2006) and Theorem 3.1 by Nanan and Dhompongsa (Fixed Point Theory Appl 54:10, 2011).