Some Negative Results in Connection with Marchaud — Type Inequalities

A classical result due to M. Eidelheit and B. Yood states that the standard algebra norm on the algebra of bounded linear operators on a Banach space is minimal, in the sense that the norm must be less than a multiple of any other submultiplicative norm on the same algebra. This definition does not assume that the arbitrary algebra norm is complete. In cases when the standard algebra norm is, in addition, maximal, it is therefore unique up to equivalence. More recently, M. Meyer showed that the Calkin algebras of a very restricted class of Banach spaces also have unique algebra norms. We generalise the Eidelheit-Yood method of proof, to show that the conventional quotient norm on a larger class of Calkin algebras is minimal. Since maximality of the norm is a presumed property for the class, the norm is also unique. We thus extend the result of Meyer. In particular, we establish that the Calkin algebras of canonical Banach spaces such as James’ space and Tsirelson’s space have unique algebra norms, without assuming completeness. We also prove uniqueness of norm for quotients of the algebras of operators on classical non-separable spaces, the closed ideals of which were previously studied by M. Daws. One aspect of the Eidelheit-Yood method is a dependence on the uniform boundedness principle. As a component of our generalisation, we prove an analogue of that principle which applies to Calkin algebra elements rather than bounded linear operators. In order to translate the uniform boundedness principle into this new setting, we take the perspective that non-compact operators map certain wellseparated sequences to other well-separated sequences. We analyse the limiting separation of such sequences, using these values to measure the non-compactness of operators and define the requisite notion of a bounded set of non-compact operators. In the cases when the underlying Banach space has a Schauder basis, we are able to restrict attention to seminormalised block basic sequences. As a consequence, our main uniqueness of norm result for Calkin algebras relies on the existence of bounded mappings between, and projections onto, the spans of block basic sequences in the relevant Banach spaces.

This chapter develops the beginnings of abstract functional analysis, a subject designed to study properties of functions by treating the functions as the members of a space and formulating the properties as properties of the space. Section 1 defines Banach spaces as complete normed linear spaces and gives a number of examples of these. The space of bounded linear operators from one normed linear space to another is a normed linear space, and it is a Banach space if the range is a Banach space. Sections 2–3 concern Hilbert spaces. These are Banach spaces whose norms are induced by inner products. Section 2 shows that closed vector subspaces of such a space have orthogonal complements, and it shows the role of orthonormal bases for such a space. Section 3 concentrates on bounded linear operators from a Hilbert space to itself and constructs the adjoint of each such operator. Sections 4–6 prove the three main abstract theorems about the norm topology of general normed linear spaces—the Hahn–Banach Theorem, the Uniform Boundedness or Banach–Steinhaus Theorem, and the Interior Mapping Principle. A number of consequences of these theorems are given. The second and third of the theorems require some hypothesis of completeness. The topic of Hilbert and Banach spaces continues in Chapter IV of the companion volume, Advanced Real Analysis.

CHAPTER 23 - Banach Spaces and Topology

The first four sections of this chapter form its core and include classical topics such as bounded linear transformations, the open mapping theorem, the closed graph theorem, the uniform boundedness principle, and the Hahn-Banach theorem. The chapter includes a good number of applications of the four fundamental theorems of functional analysis. Sections 6.5 and 6.6 provide a good account of the properties of the spectrum and adjoint operators on Banach spaces. They may be largely bypassed, since the treatment of the corresponding topics for operators on Hilbert spaces in chapter 7 is self-contained. The section on weak topologies is more advanced and may be omitted if a brief introduction is the goal. The chapter is enriched by such topics as the best polynomial approximation, the Hilbert cube, Gelfand’s theorem, Schauder bases, complemented subspaces, and the Banach-Alaoglu theorem.

In this paper, we establish some new characterizations of a weight \(w\) such that discrete Hardy operator \(\mathcal{H}f(n):=\frac{1}{n}\sum\limits_{s=1}^{n}f(s)\) for quasi-nonincreasing sequence \(f(n)\) is bounded in the Banach space \(\ell _{w}^{p}(\mathbb{Z}_{+})\) when \(0<p<\infty .\) In particular, we will prove that \(\mathcal{H}f\) is bounded in \(\ell _{w}^{p}(\mathbb{Z}_{+})\) if and only if \(w\) belongs to the \(\beta\)-discrete Arino and Muckenhoupt class \(\mathcal{B}_{p,\beta }\). We prove that the self-improving property for the class \(\mathcal{B}_{p,\beta }\) holds, that is we prove that if \(w\in \mathcal{B}_{p,\beta }\) then there exists an \(\varepsilon >0\) such that \(w\in \mathcal{B}_{p-\varepsilon ,\beta }.\)

Hyperbolic hemivariational inequalities controlled by evolution equations with application to adhesive contact model

In this paper, we establish the sharp Fekete–Szegö inequalities of functional type over a new class of holomorphic mappings on the unit ball in a complex Banach space. This class includes some important widely studied classes. We offer simpler proofs and require less restrictive assumptions than those found in recent literature. Finally we show that some important previous results follows from ours by choosing suitable parameters.

Defect of compactness, relative to an embedding of two Banach spaces E and F, is a difference between a weakly convergent sequence in E and its weak limit taken up to a remainder that vanishes in the norm of F. For Sobolev embeddings in particular, defect of compactness is expressed as a profile decomposition - a sum of terms, called elementary concentrations, with asymptotically disjoint supports. We discuss a profile decomposition for the Sobolev space of a Riemannian manifold with bounded geometry, which is a sum of elementary concentrations associated with concentration profiles defined on manifolds different from M, that are induced by a limiting procedure. The profiles satisfy an inequality of Plancherel type, and a similar relation, related to the Brezis-Lieb Lemma, holds for Lebesgue norms of profiles on the respective manifolds.

In this chapter, we continue our study of Banach and Hilbert spaces. Here, we mainly consider linear functionals, i.e., additive, homogeneous, and continuous number functions given on such spaces. The problems under consideration are mostly grouped around two fundamental facts, namely, the Hahn-Banach theorem on extensions of linear functionals and the Banach-Steinhaus theorem (the principle of uniform boundedness). The general form of linear continuous functionals in many important spaces and some geometric problems in the theory of Hilbert spaces are also investigated.

For vector lattices E and F, where F is Dedekind complete and supplied with a locally solid topology, we introduce the corresponding locally solid absolute strong operator topology on the order bounded operators {mathscr{L}}_{mathrm{ob}}(E,F) from E into F. Using this, it follows that {mathscr{L}}_{mathrm{ob}}(E,F) admits a Hausdorff uo-Lebesgue topology whenever F does. For each of order convergence, unbounded order convergence, and—when applicable—convergence in the Hausdorff uo-Lebesgue topology, there are both a uniform and a strong convergence structure on {mathscr{L}}_{mathrm{ob}}(E,F). Of the six conceivable inclusions within these three pairs, only one is generally valid. On the orthomorphisms of a Dedekind complete vector lattice, however, five are generally valid, and the sixth is valid for order bounded nets. The latter condition is redundant in the case of sequences of orthomorphisms, as a consequence of a uniform order boundedness principle for orthomorphisms that we establish. We furthermore show that, in contrast to general order bounded operators, orthomorphisms preserve not only order convergence of nets, but unbounded order convergence and—when applicable—convergence in the Hausdorff uo-Lebesgue topology as well.

The purpose of this paper is to indicate a unified approach to quantitative negative results in numerical analysis. This is done via a rather general theorem which in fact subsumes our previous quantitative uniform boundedness principles. The proof is based upon a gliding hump method. The general theorem is exemplarily applied to discuss the sharpness of various direct and inverse approximation results, known for the compound trapezoidal rule and for the approximate solution of the heat equation. The treatment outlines a program which may also be worked out for other procedures.

A Banach space X is said to be sequentially unconditional if every Schauder basic sequence has an unconditional subsequence. We provide an example of a reflexive Banach space Xisu which is sequentially unconditional and such that every T ∈ L(Xisu) is a strictly singular perturbation of a multiple of the identity. The space Xisu belongs to the class of Banach spaces with saturated and conditional structure. Constructing Xisu we follow the general scheme invented by W. T. Gowers and B. Maurey in their celebrated construction of a Hereditarily Indecomposable Banach space. The new ingredient concerns the definition of the special (conditional) functionals. For this we use a non-injective coding function σ in conjunction with a Baire-like property of a sequence of countable trees stated and proved in the present paper. It is worth mentioning that Xisu admits a richer unconditional structure than classical spaces like L1(0,1) and it remains unsettled whether Xisu is a subspace of a Banach space with an unconditional basis. 2000 Mathematics Subject Classification 46B20, 46B45 (primary), 05D99, 47L05 (secondary).

A variation on the uniform boundedness theorem [2, p. 105] of functional analysis is obtained by replacing pointwise boundedness by a slightly stronger type of boundedness. This makes it possible to remove the category requirements from the set over which it is desired to have a collection of functions uniformly bounded. The concept of boundedness to be utilized is called boundedness on Cauchy sequences (see Definition 1). When a Cauchy sequence converges in the domain of a family of continuous functions, boundedness on a Cauchy sequence is equivalent to pointwise boundedness of the family at the limit point of the point of the sequence. When the domain of the continuous functions is a metric space, boundedness on the collection on all Cauchy sequences is equivalent to pointwise boundedness of the continuous extensions on the completion of the domain. The interesting case is when the domain is not metrizable because boundedness on Cauchy sequences does not necessarily imply pointwise boundedness of the extended functions on the completion or even on the sequential completion of the domain. The main result of the paper is given in Theorem 1 which is followed by six corollaries. The sixth corollary is especially interesting in stating that the closure of an infrabarrelled space in its second adjoint space is a barreled space. The theorem on uniform boundedness immediately suggests an absorption theorem which is Theorem 2 of the paper. Theorem 1 does not follow as a corollary of Theorem 2 as is the case of the classical uniform boundedness theorem [2, p. 105]. DEFINITION 1. A family H of continuous linear transformations with common domain a linear topological space E and ranges in a locally convex space F is bounded on a Cauchy sequence {Xn } in E if for every neighborhood V of the origin in F there exists a positive real number r with the property that for each f in H there is an Nf such that f (xn) is in r V for all n > Nf. The following theorem is stated in the context of the definition.

The aim of the present paper is to establish some new integral inequalities of Opial type involving functions of three independent variables and their partial derivatives. Our results yields in the special cases some of the inequalities recently appeared in the literature.

Infinite dimensional subspaces, which consists of forms involving tensor products of functions of fewer variables, play an important role in the theory of approximation of functions of several real variables. Similar technique can be applied to problems of approximation of analytic functions of several complex variables. In the present paper we study the generalized polynomial approximation of entire functions of two complex variables in Banach spaces. Moreover, limit equations relating growth parameters order and type with above approximation errors in Banach spaces have been investigated.