Isometries on Banach Spaces

Isometries on Banach Spaces : Function Spaces

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Fundamental to the study of any mathematical structure is an understanding of its symmetries. In the class of Banach spaces, this leads naturally to a study of isometries-the linear transformations that preserve distances. In his foundational treatise, Banach showed that every linear isometry on the space of continuous functions on a compact metric space must transform a continuous function x into a continuous function y satisfying y(t) = h(t)x(p(t)), where p is a homeomorphism and |h| is identically one. Isometries on Banach Spaces: Function Spaces is the first of two planned volumes that survey investigations of Banach-space isometries. This volume emphasizes the characterization of isometries and focuses on establishing the type of explicit, canonical form given above in a variety of settings. After an introductory discussion of isometries in general, four chapters are devoted to describing the isometries on classical function spaces. The final chapter explores isometries on Banach algebras. This treatment provides a clear account of historically important results, exposes the principal methods of attack, and includes some results that are more recent and some that are lesser known. Unique in its focus, this book will prove useful for experts as well as beginners in the field and for those who simply want to acquaint themselves with this area of Banach space more

Product details

  • Hardback | 208 pages
  • 161.5 x 240.8 x 16.8mm | 444.53g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 1584880406
  • 9781584880400

Review quote

"This is a very well written book. The authors have done a remarkable job in collecting this material and in exposing it in a very clear style. It will be an important reference tool for analysts, experts and nonexperts, and it will provide a clear and direct path to several topics of current research interest." - Juan J. Font, Mathematical Reviews, Issue 2004jshow more

Table of contents

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 INDEXshow more