A Characterization of l + 1(Γ)

CHAPTER 23 - Banach Spaces and Topology

It is shown that there is a nonseparable Banach space on which every operator has the form S + pi where p is a scalar and S is an operator with separable range.An open problem in the theory of Banach spaces is the construction of a Banach space with the property that all operators from it to itself are the sum of a multiple of the identity and a compact operator.Since compact operators have separable range a related question is to find a (nonseparable) Banach space on which all operators are the sum of a multiple of the identity and an operator with separable range.This was first done by Shelah using an extra set-theoretic axiom in [1].It is the purpose of this paper, which is the continuation of research which began in [3], to prove the same result without appealing to axioms other than those of ZFC.The following notation will be used.The integers will be considered as ordinals so that n Gm means the same as n < m.The set of integers is denoted as w instead of N and the first uncountable ordinal is wi.The symbols [X]x and [X]<A denote, respectively, the set of A sized subsets of X and less than A sized subsets of X.The set of functions from Y to X is denoted YX.In particular, WlR, when endowed with the usual pointwise operations, is the real vector space of "wi-tuples".For X ui the characteristic function of X will be denoted by lx-Let us begin by reviewing the construction from [2] of a Banach space on which every operator is the sum of an operator with separable range and one which is diagonal with respect to a fixed basis.Let 21 [u>i]<N be such that:i. ua = wi.2. 21 is closed under subsets.It is possible to define a function || ||a on ">R by ||/||*=sup{{|/(a)|;aGA};AG2l}.Let B = {UlR; ||/||a is finite} and note that (B, || ||a) is a normed linear space.Of course if 21 = [wi]<"' then B = /i(wi) but we will assume that there is a family 53 [wi]<w such that: 3. 93 is closed under subsets.4. an8c [uji]1.5.For all families of disjoint pairs {{a(a),b(a)}; a G u>i} and integers k there is X G [ui]k such that {a(a); a G X} G 21 and {6(a); a G X} G 03.

This paper gives a necessary and sufficient condition in order that a bounded linear mapping from L 1 ( μ ) {L^1}(\mu ) into a Banach space be compact. It is applied to provide a slightly improved form of the Radon-Nikoydm theorem for vector valued measures and to give a sufficient condition in order that the range of a vector valued measure of bounded variation be compact.

In Section 4.7 we introduced the space L p (µ) for p ≥ 1 and mentioned that they are Banach spaces. Among these the space L2(µ) has a special role to play in our subject. In fact all statistical problems which centre around ‘extrapolation’ or ‘prediction’ are based on the fundamental properties of the space L2(µ). To elaborate on this theme we shall develop the barest minimum of the theory of Banach and Hilbert spaces.

This dissertation is concerned with the interplay between the theory of Banach spaces and representations of groups. The wavelet transform has proven to be a useful tool in characterizing and constructing Banach spaces, and we investigate a generalization of an already known technique due to H.G. Feichtinger and K. Gröchenig. This generalization is presented in Chapter 3, and in Chapters 4 and 5 we present examples of spaces which can be described using the theory. The first example clears up a question regarding a wavelet characterization of Bergman spaces related to a non-integrable representation. The second example is a wavelet characterization of Besov spaces on the forward light cone.

The publication of Banach's book [2] in 1932 might be regarded as marking the beginning of the systematic study of Banach spaces. Research activity in this area has expanded dramatically during the past two decades. Interesting new directions have developed and interplays between Banach space theory and other mathematics have proved to be very valuable. Most well-known classical problems have been solved, but some remain and important new problems have arisen. Our purpose is very limited. It is to discuss some of the most important and fundamental facts about bases in Banach spaces, particularly those that may be interesting and useful for mathematicians in other fields. Often, only sketches of proofs will be given, while others are given in more detail because they are relatively easy and may contribute to developing intuitive feeling for the concepts involved. It will be seen that many very important and beautiful theorems have very easy and natural proofs. A basis (or Schauder basis) for a Banach space X is a sequence (en: n > 1) of members of X which has the property that, for each x in X, there is exactly one sequence of scalars (xi} for which x = xiei in the sense that lim.nX lix - E ixieijj = 0. Some important Banach spaces have very natural bases. If (en} is a complete orthonormal sequence in Hilbert space H and x is any member of H, then there is exactly one sequence (xn} of scalars such that lim x yn- lxieill = 0. For this sequence of scalars, each xi is (x, ei) and n 00 ~~~~~1/2

Milutin's Theorem states that if X and Y are uncountable metrizable compact Hausdorff spaces, then C(X) and C(Y) are isomorphic as Banach spaces [15, p. 379]. Thus there is only one isomorphism class of such Banach spaces. There is also an extensive theory of the Banach–Mazur distance between various classes of classical Banach spaces with the deepest results depending on probabilistic and asymptotic estimates [18]. Lindenstrauss, Haagerup and possibly others know that as Banach spaceswhere H is the infinite dimensional separable Hilbert space, R is the injective II 1-factor on H, and ≈ denotes Banach space isomorphism. Haagerup informed us of this result, and suggested considering completely bounded isomorphisms; it is a pleasure to acknowledge his suggestion. We replace Banach space isomorphisms by completely bounded isomorphisms that preserve the linear structure and involution, but not the product. One of the two theorems of this paper is a strengthened version of the above result: if N is an injective von Neumann algebra with separable predual and not finite type I of bounded degree, then N is completely boundedly isomorphic to B(H). The methods used are similar to those in Banach space theory with complete boundedness needing a little care at various points in the argument. Extensive use is made of the conditional expectation available for injective algebras, and the methods do not apply to the interesting problems of completely bounded isomorphisms of non-injective von Neumann algebras (see [4] for a study of the completely bounded approximation property).

In his book Transversal Theory (4) Mirsky introduces the problem of finding a necessary and sufficient condition in order that a finite family of sets have a unique independent transversal. The purpose of this note is to give one necessary and sufficient condition. More generally, we give a necessary and sufficient condition in order that a common basis of two rank-finite matroids be unique. We then specialize this result to give an answer to the above question.

In recent years the use of the techniques from the geometric theory of Banach spaces in the study of function spaces has become an important tool and we are happy that this special issue has received the attention of both senior and young researchers in these areas. We have received papers from a wide cross-section of researchers in this area, covering existence theorems concerning the convolution of functions, distributions, and ultradistributions of Beurling type with supports satisfying suitable compatibility conditions and the Rayleigh equation of two deviating arguments. Application of the Leray-Schauder index theorem and the Leray-Schauder fixed point theorem towards the existence of nontrivial periodic solutions of this type of equations is covered. Also covered is smooth approximation of Lipschitz functions on Finsler manifolds, keeping control on the corresponding Lipschitz constants. An article that extends the usual notion ofKothe dual of a Banach function space to a broad class of Banach lattices is also in this collection. On abstract Banach space theory, there is an article that studies a weakening of the notion of central subspace of a Banach space, as well as an article that studies notions of generalized approximation properties.

&lt;p&gt;&lt;b&gt;We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.&lt;/b&gt;&lt;/p&gt; &lt;p&gt;We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.&lt;/p&gt; &lt;p&gt;However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.&lt;/p&gt; &lt;p&gt;We also investigate the effectivisations of certain classical theorems in Banachspaces.&lt;/p&gt;

In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an Lp-space, then it is either an Lp-space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non- Hilbertian complemented operator subspaces of non-commutative Lp-spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of operator spaces for every 2 < p < 1 which can be considered as operator space analogues of the Rosenthal sequence spaces from Banach space theory, constructed in 1970. Under the usual conditions on the defining sequence we prove that most of these spaces are operator Lp-spaces, not completely isomorphic to previously known such spaces. However, it turns out that some column and row versions of our spaces are not operator Lp-spaces and have a rather complicated local structure which implies that the Lindenstrauss-Rosenthal alternative does not carry over to the non-commutative case.

The article is a survey related to a classical unsolved problem in Banach space theory, appearing in Banach’s famous book in 1932, and known as the Mazur rotations problem. Although the problem seems very difficult and rather abstract, its study sheds new light on the importance of norm symmetries of a Banach space, demonstrating sometimes unexpected connections with renorming theory and differentiability in functional analysis, with topological group theory and the theory of representations, with the area of amenability, with Fraïssé theory and Ramsey theory, and led to development of concepts of interest independent of Mazur problem. This survey focuses on results that have been published after 2000, stressing two lines of research which were developed in the last 10 years. The first one is the study of approximate versions of Mazur rotations problem in its various aspects, most specifically in the case of the Lebesgue spaces \(L_p\). The second one concerns recent developments of multidimensional formulations of Mazur rotations problem and associated results. Some new results are also included.

This textbook is an introduction to the techniques of summing and nuclear norms. The author's aim is to present a clear and simple account of these ideas and to demonstrate the power of their application to a variety of Banach space questions. The style is expository and the only prerequisite is a beginner's course on Wormed linear spaces and a minimal knowledge of functional analysis. Thus, Dr Jameson is able to concentrate on important, central results and gives concrete and largely non-technical proofs, often supplying alternative proofs which both contribute something to the understanding. Final-year undergraduates and postgraduates in functional analysis will enjoy this introduction to the subject, and there are many examples and exercises throughout the text to help the reader and to demonstrate the range of application these techniques find. A list of references indicates the way for further reading.

In this paper we provide sufficient conditions in order that Uryson's equation has a nontrivial solution. Using a modified version of. Krasnoselskii's fixed point theorem on operators expanding a cone in a Banach space we get two new existence results for superlinear equations. The special structure of cones used provides bounds of the solution.

Let E be a real Banach space, C a closed, convex subset of E and f : [ 0 , 1 ] × E × E → E f:[0,1] \times E \times E \to E be continuous. Let u 0 , u 1 ∈ C {u_0},{u_1} \in C and consider the boundary value problem ( ∗ ) u = f ( t , u , u ′ ) , u ( 0 ) = u 0 , u ( 1 ) = u 1 . \begin{equation}\tag {$\ast $} u = f(t,u,u’),\quad u(0) = {u_0},\quad u(1) = {u_1}.\end{equation} We establish sufficient conditions in order that ( ∗ ) (\ast ) have a solution u : [ 0 , 1 ] → C u:[0,1] \to C .