Representations of Linear Operators Between Banach Spaces

1. Front Matter

   Pages i-xi

   PDF

2. Preliminaries

   • David E. Edmunds, W. Desmond Evans

   Pages 1-66

3. Representation of Compact Linear Operators

   • David E. Edmunds, W. Desmond Evans

   Pages 67-126

4. Representation of Bounded Linear Operators

   • David E. Edmunds, W. Desmond Evans

   Pages 127-140

5. Back Matter

   Pages 141-152

   PDF

The book deals with the representation in series form of compact linear operators acting between Banach spaces, and provides an analogue of the classical Hilbert space results of this nature that have their roots in the work of D. Hilbert, F. Riesz and E. Schmidt. The representation involves a recursively obtained sequence of points on the unit sphere of the initial space and a corresponding sequence of positive numbers that correspond to the eigenvectors and eigenvalues of the map in the Hilbert space case. The lack of orthogonality is partially compensated by the systematic use of polar sets. There are applications to the p-Laplacian and similar nonlinear partial differential equations. Preliminary material is presented in the first chapter, the main results being established in Chapter 2. The final chapter is devoted to the problems encountered when trying to represent non-compact maps.

• approximation property
• compact operators
• p-Laplacian
• strictly and unformly convex Banach spaces
• partial differential equations

From the reviews:

“Book presents an account of the spectral theory of operators acting in Banach spaces. … notes and comments at the end of each chapter give a fairly complete documentation, enabling the reader to trace the material to its sources, pursue the topics further and see them in context. As an authoritative account of a new and rapidly developing branch of spectral theory, this work will be of great interest to research workers and students in the field and related topics.” (Petru A. Cojuhari, zbMATH, Vol. 1283, 2014)

• Department of Mathematics, University of Sussex, Brighton, United Kingdom

  David E. Edmunds

• School of Mathematics, University of Cardiff, Cardiff, United Kingdom

  W. Desmond Evans

Policies and ethics