In this tract, Dr Ruston presents analogues for operators on Banach spaces of Fredholm's solution of integral equations of the second kind. Much of the presentation is based on research carried out over the last twenty-five years and has never appeared in book form before. Dr Ruston begins with the construction for operators of finite rank, using Fredholm's original method as a guide. He then considers formulae that have structure similar to those obtained by Fredholm, using, and developing further, the relationship with Riesz theory. In particular, he obtains bases for the finite-dimensional subspaces figuring in the Riesz theory. Finally he returns to the study of specific constructions for various classes of operators. Dr Ruston has made every effort to keep the presentation as elementary as possible, using arguments that do not require a very advanced background. Thus the book can be read with profit by graduate students as well as specialists working in the general area of functional analysis and its applications.
The first edition of this well known book was noted for its clear and accessible exposition of the basic theory of Hardy spaces from the concrete point of view (in the unit circle and the half plane). The intention was to give the reader, assumed to know basic real and complex variable theory and a little functional analysis, a secure foothold in the basic theory, and to understand its applications in other areas. For this reason, emphasis is placed on methods and the ideas behind them rather than on the accumulation of as many results as possible. The second edition retains that intention, but the coverage has been extended. The author has included two appendices by V. P. Havin, on Peter Jones' interpolation formula, and Havin's own proof of the weak sequential completeness of L1/H1(0); in addition, numerous amendments, additions and corrections have been made throughout.
1. Essentials 2. Restricted linear spaces 3. Line sizes 4. Semi-affine linear spaces 5. Complementation 6. General configurations 7. n-Dimensional linear spaces 8. Groups Appendix.
This book presents an overview of modern Banach space theory. It contains sixteen papers that reflect the wide expanse of the subject. Articles are gathered into five sections according to methodology rather than the topics considered. The sections are: geometrical methods; homological methods; topological methods; operator theoretic methods; and also function space methods. Each section contains survey and research papers describing the state-of-the-art in the topic considered as well as some of the latest most important results. Researchers working in Banach space theory, functional analysis or operator theory will find much of interest here.
Topological and Metric Spaces Banach Spaces Bounded Operators Hilbert Spaces Operators in Hilbert Space Spectral Theory Integral Operators Semigroups of Evolution Sobolev Spaces Interpolation Spaces Linear Elliptic Operators Regularity of Hyperbolic Mixed Problems The Hilbert Uniqueness Method Exercises References
A continuation of the authors' previous book, Isometries on Banach Spaces: Vector-valued Function Spaces and Operator Spaces, Volume Two covers much of the work that has been done on characterizing isometries on various Banach spaces. Picking up where the first volume left off, the book begins with a chapter on the Banach-Stone property.
BEGINNINGS Introduction Banach's Characterization of Isometries on C(Q) The Mazur-Ulam Theorem Orthogonality The Wold Decomposition Notes and Remarks CONTINUOUS FUNCTION SPACES--THE BANACK-STONE THEOREM Introduction Eilenberg's Theorem The Nonsurjective case A Theorem of Vesentini Notes and Remarks THE L(p) SPACES Introduction Lamperti's Results Subspaces of L(p) and the Extension Theorem Bochner Kernels Notes and Remarks ISOMETRIES OF SPACES OF ANALYTIC FUNCTIONS Introduction Isometries of the Hardy Spaces of the disk Bergman spaces Bloch Spaces S(p) Spaces Notes and Remarks REARRANGEMENT INVARIANT SPACES Introduction Lumer's Method for Orlicz Spaces Zaidenberg's Generalization Musielak-Orlicz Spaces Notes and Remarks BANACH ALGEBRAS Introduction Kadison's Theorem Subdifferentiability and Kadison's Theorem The Nonsurjective Case of Kadison's theorem The Algebras C(1) and AC Douglas Algebras Notes and Remarks BIBLIOGRAPHY INDEX
PART I INTRODUCTION TO BANACH SPACES 1. Preliminaries 2. Elements of normed spaces 3. Banach spaces PART II BANACH ALGEBRAS 4. Banach algebras 5. Representation theory 6. Algebras with an involution 7. The Borel functional calculus PART III SCV AND BANACH ALGEBRAS 8. Introduction to several complex variables 9. The holomorphic functional calculus in several variables Bibliography Index
This book presents a comprehensive treatment of aspects of classical and modern analysis relating to theory of âpartial differential equationsâ and the associated âfunction spacesâ. It begins with a quick review of basic properties of harmonic functions and Poisson integrals and then moves into a detailed study of Hardy spaces. The classical Dirichlet problem is considered and a variety of methods for its resolution ranging from potential theoretic (Perronâs method of sub-harmonic functions and Wienerâs criterion, Greenâs functions and Poisson integrals, the method of layered potentials or integral equations) to variational (Dirichlet principle) are presented. Parallel to this is the development of the necessary function spaces: Lorentz and Marcinkiewicz spaces, Sobolev spaces (integer as well as fractional order), Hardy spaces, the John-Nirenberg space BMO, Morrey and Campanato spaces, Besov spaces and Triebel-Lizorkin spaces. Harmonic analysis is deeply intertwined with the topics covered and the subjects of summability methods, Tauberian theorems, convolution algebras, Calderon-Zygmund theory of singular integrals and Littlewood-Paley theory that on the one hand connect to various PDE estimates (Calderon-Zygmund inequality, Strichartz estimates, Mihlin-Hormander multipliers, etc.) and on the other lead to a unified characterisation of various function spaces are discussed in great depth. The book ends by a discussion of regularity theory for second order elliptic equations in divergence formâ first with continuous and next with measurable coefficientsâand covers, in particular, De Giorgiâs theorem, Moser iteration, Harnack inequality and local boundedness of solutions. (The case of elliptic systems and related topics is discussed in the exercises.)
This monograph presents recent developments in spectral conditions for the existence of periodic and almost periodic solutions of inhomogenous equations in Banach Spaces. Many of the results represent significant advances in this area. In particular, the authors systematically present a new approach based on the so-called evolution semigroups with
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