Structure Of Banach Spaces Research Articles

Iterative Methods Applied to Matrix Equations Found in Calculating Spheroidal Functions

We look at iterative methods for solving matrix equations, particularly those matrices with small entries. Iterative methods aid computational stability by relying on the topological structure of Banach or Hilbert spaces rather than depending on a calculation's numerical precision. When applicable, they are also quicker than Gaussian elimination. As an example, we use these methods to tabulate the expansion of periodic spheroidal functions in associated Legendre functions, given arbitrary values of the parameters appearing in its defining differential equation. These functions appear in solutions to 3-D Helmholtz equations in oblate and prolate spheroidal coordinates as well as a 1-D Schrödinger equation.

Regularity of solutions to the perturbed conservation laws

A kind of regularity for the mild solution of perturbed conservation laws is proposed. This regularity is described in term of variations measured in the L1-norm. A dissipativity condition from the semigroup approach is used to show that the mild solution stays within a class of bounded variation in this sense of regularity. This shows that this class of functions is an invariant of the semigroup. The same analysis carries over to the periodic problem. The class of boundedL1-variation functions used here can be normed to give a Banach space structure. It also has an analogue with the space of Lipschitz functions

Fourier–Stieltjes Algebras of Locally Compact Groupoids

For locally compact groups, Fourier algebras and Fourier–Stieltjes algebras have proven to be useful dual objects. They encode the representation theory of the group via the positive definite functions on the group: positive definite functions correspond to cyclic representations and span these algebras as linear spaces. They encode information about the algebra of the group in the geometry of the Banach space structure, and the group appears as a topological subspace of the maximal ideal space of the algebra. Because groupoids and their representations appear in studying operator algebras, ergodic theory, geometry, and the representation theory of groups, it would be useful to have a duality theory for them. This paper gives a first step toward extending the theory of Fourier–Stieltjes algebras from groups to groupoids. IfGis a locally compact (second countable) groupoid, we show that B(G), the linear span of the Borel positive definite functions onG, is a Banach algebra when represented as an algebra of completely bounded maps on aC*-algebra associated withG. This necessarily involves identifying equivalent elements of B(G). An example shows that the linear span of the continuous positive definite functions need not be complete. For groups, B(G) is isometric to the Banach space dual ofC*(G). For groupoids, the best analog of that fact is to be found in a representation of B(G) as a Banach space of completely bounded maps from aC*-algebra associated withGto aC*-algebra associated with the equivalence relation induced byG. This paper adds weight to the clues in earlier work of M. E. Walter on Fourier–Stieltjes algebras that there is a much more general kind of duality for Banach algebras waiting to be explored.

Discovering the algebraic structure on the metric injective envelope of a real Banach space

We exhibit explicitly the structure of Banach space on the metric injective envelope of a real Banach space, in a simple and unexpected way.

Property (M), M-ideals, and almost isometric structure of Banach spaces.

Property (M), M-ideals, and almost isometric structure of Banach spaces.

Properties (V) and (u) are not three-space properties

In his fundamental papers [7,8], Pelczynski introduced properties (u), (V), and (V*) as tools as study the structure of Banach spaces. Let X be a Banach space. It is said that X has property (u) if, for every weak Cauchy sequence (xn) in X, there exists a weakly unconditionally Cauchy (wuC) series in X such that the sequence is weakly null. It is said that X has property (V) if, for every Banach space Z, every unconditionally converging operator from X into Z is weakly compact; equivalently, whenever K is a bounded subset of X* such that for every wuC series in X, then K is relatively weakly compact. A Banach space X is said to have property (V*) if whenever K is a bounded subset of X such that 0 for every wuC series in X*, then K is relatively weakly compact. Some well-known results which shall be needed later are contained in the following.

The Functional Analysis of the X-Ray Transform on Trees

The X-ray transform plays an important role in the integral geometry of trees. It transforms functions on the vertices into functions on the geodesics. In (Casadio Tarabusi, Cohen, and Picardello, Adv. in Math., to appear) a precise description of the domain and range of the X-ray transform was given. In this work we construct and analyze subsets of these function spaces which have a Banach space structure and on which the X-ray transform acts as an isometry. Questions concerning continuity on the space of geodesics are also studied.

Beurling’s analyticity theorem

In 1967, I sent some reprints to the Institute for Advanced Study as part of an application for a visiting membership. In one of these papers [5], Arne Beurling saw a result of mine which suggested to him a conjecture which he subsequently proved. The result is still scarcely known. It is a real variable result whose proof uses some complex analysis and harmonic analysis. It has implications for the study of one-parameter semigroups of operators, Markov processes, structure of Banach spaces, and certain aspects of chaos theory. It is the intent of this article to describe the result, give the setting under which it was found, and indicate some of its implications. I also hope to contribute to a fuller picture of Beufling. First, I describe what Beurling saw which led to the discovery of his theorem. We need some notation for higher-order finite differences: There is the following easy consequence:

Some properties of monotone type multivalued operators in Banach spaces

Some properties of monotone type multivalued operators including accretive operators and the duality mapping are studied in connection with the structure of Banach spaces.

Approximation Theory and Functional Analysis.

Part 1 Research and survey articles: on Bernstein-Durrineyer polynomials with Jacob weights, H.Berens and Y.Xu an overview of wavelets, C.Chui a note on weak inequalities in Orlicz and Lorentz spaces, G.A.Edgar and L.Sucheston bivariate Birkhoff interpolation - a survey, R.A.Lorentz linear approximations of functions with several restricted derivatives, Y.Makovoz some characterizations theorems for measures associated with orthogonal polynomials on the unit circle, K.Pan and E.B.Saff box splines, cardinal series, and wavelets, S.D.Reimenschneider and Z.Shen some aspects of the subspace structure of infinite dimensional banach space, H.Rosenthal fairness and monotone curvature, J.Roulier et al projections on 2-dimensional spaces, N.Tomczak-Jacgermann real versus complex best rational approximation, R.S.Varga and A.Ruttan.

SCHAUDER BASIS DETERMINING PROPERTIES

In this paper we introduce a new concept, called the Schauder basis determining property, which is strictly stronger than the concept of a countably determining property. This concept provides a new way to study the structure of Banach spaces, especially, the goometrical properties of Banach spaces. We show that many structural properties of Banach spaces are Schauder basis determining properties. These include the Banach-Saks property, the Asplund property, full convexity, the Kadee-Klee property, reflexity, some-what refiexity, property (A) — (E), the Schur property, and the Krein-Milman property. We also give a counterexample to show that the approximation property is not a Schauder basis determining property, but it is a countably determining property.

On the structure of Banach spaces with Mazur's intersection property

On the structure of Banach spaces with Mazur's intersection property

On almost symmetric sequences inL p

The case p--1 in Theorem 1 is due to H. P. RosenthaI (unpublished) and the cases p =2, 4, 6 . . . . to W. B. Johnson, B. Maurey, G. Schechtman and L. Tzafriri [91. The notion of almost symmetric sequences comes from functional analysis; we refer to [7] and [9] for applications of almost symmetry and Theorem 1 for the study of the structure of Banach spaces. In [8] S. Guerre and Y. Raynaud showed that for every 1-<p<2 there exists a weakly null sequence in Lp without almost symmetric subsequences. Hence condition (2) in Theorem 1, although it can be considerably weakened (see the Corollary after Theorem 2) cannot be omitted completely. The example in [8] also shows that despite the strong similarity between sparse subsequences of Lp-bounded sequences and exchangeable sequences (see Aldous [1]), not all asymptotic properties of sparse sequences are symmetric. No characterization of sequences (X,) having an almost symmetric subsequence is known.

Some affine geometric aspects of operator algebras

The predual of a von Neumann algebra is shown to be a neutral strongly facially symmetric space, thereby suggesting an affine geometric approach to operator algebras and their non-associative analogues. Geometric proofs are obtained for the polar decompositions of normal functionals in ordered and non-ordered settings. A fundamental problem in the operator algebraic approach to quantum mechanics is to determine those algebraic structures in Banach spaces which are characterized by a set of geometrical axioms defining the quantum mechanical measuring process. This problem was solved in the context of ordered Banach spaces by Alfsen, Hanche-Olsen, and Shultz ([2], [1]) and led to a characterization of the state spaces of /2?*-algebras and C*-algebras. The main thrust of the present authors' recent research has been to find those algebraic structures induced on (unordered) Banach spaces in which such quantum mechanical axioms are satisfied. This project, which was initiated in [14] and [15] using the affine geometry of the dual unit ball, is used here to give a geometric proof of the Tomita-Sakai-Effros polar decomposition of a normal functional on a von Neumann algebra. Thus, the purpose of this partially expository paper is to show the richness and power of the affine geometric structure of the dual space of an operator algebra, by working in a purely geometric model. Indeed, since this geometry can be described in terms of the underlying real structure, it can be used to obtain new results in the real structure of operator algebras and in the structure of real operator algebras. For example, by using this approach, Dang ([5]) has shown that a raz/-linear isometry of a C*-algebra is the sum of a linear and a conjugate linear isometry, and hence is multiplicative, thereby obtaining a real analogue of Kadison's non-commutative extension of the BanachStone Theorem. The category of strongly facially symmetric (SFS) spaces (simply called facially symmetric spaces in [14] and [15]) has been shown to

Homomorphisms Between Algebras of Continuous Functions

We are concerned in this paper with the study of homomorphisms between different algebras of continuous functions, especially the algebras of real functions which are either weakly continuous on bounded sets or weakly uniformly continuous on bounded sets on a Banach space (see definitions below).These spaces of weakly [uniformly] continuous functions appeared in relation with some questions in Infinite-dimensional Approximation Theory (see [4], [6], [11], [12], [13] and [16]); and since the structure of these function spaces is closely related with properties of different weak topologies (the bounded-weak and bounded-weak* topologies, respectively) and with the structure of Banach spaces on which they are defined, their study also presents interest from the point of view of Banach space theory, as can be seen in [2], [12] or [17].

Some topological and geometrical structures in Banach spaces

Some topological and geometrical structures in Banach spaces

A superreflexive Banach space which does not admit complex structure

We construct an infinite-dimensional superreflexive real Banach space which does not admit complex structure and consequently is not isomorphic to the Cartesian square of any Banach space. We also construct a variant of Bourgain’s example of a complex Banach space with nonunique complex structure and state a number of open problems about structure of Banach spaces and their linear groups.

On the structure of superfields in a field theory on a supermanifold

In view of applications to the formulation of gauge field theories on supermanifolds, we study the relation between the sheaves of functions on a supermanifold M and its body manifold M0, respectively. The nonuniqueness of the local injections t: M0→M is analysed in consideration of its role in supersymmetric field theories. A Banach space structure is given to the set of bounded, supersmooth, Ckfields on M in order to get a rigorous formulation of variational principles for the class of theories under consideration.

Geometry in quotient reflexive spaces

AbstractThe structure and geometry of Banach spaces with the property that E(4) = Ê** + E⊥ ⊥ are investigated: such spaces are called quotient reflexive spaces here. For these spaces, if E is very smooth, Ê is also very smooth, and if E* is weakly locally uniformly rotund (WLUR), E(4) is smooth on a certain (relatively) norm dense subset of Ê**. Consequently, for quotient reflexive spaces, WLUR and very-WLUR are equivalent in E*.

Bifurcating Instability of the Free Surface of a Ferrofluid

Consider a slab of ferrofluid bounded below by a fixed boundary and above by a vacuum. If the fluid is subjected to a vertically directed magnetic field of sufficient strength, surface waves appear. The equations which describe this phenomenon are derived. In the physical space no natural Banach space structure is available due to the free surface. In order to use the available bifurcation theory, a transformation of coordinates is made, mapping the surface flat. In the new coordinate system the equations define an operator between Banach spaces. The minimum eigenvalue of the linearized operator is the critical magnetic field strength where the planar surface loses stability. Using a generalized inverse of the Fréchet derivative of the operator and the implicit function theorem, the existence of a nontrivial branch of solutions is proved. A local stability criterion is also obtained and applied to three periodic structures: rolls, squares and hexagons.