Spectral theory of linear differential operators and comparison algebras / H.O. Cordes.

Format:

Book

Author/Creator:

Cordes, H. O. (Heinz Otto), 1925- author.

Series:

London Mathematical Society lecture note series ; 76.

London Mathematical Society lecture note series ; 76

Language:

English

Subjects (All):

Differential operators.

Linear operators.

Physical Description:

1 online resource (ix, 342 pages) : digital, PDF file(s).

Other Title:

Spectral Theory of Linear Differential Operators & Comparison Algebras

Place of Publication:

Cambridge : Cambridge University Press, 1987.

Language Note:

English

Summary:

The main aim of this book is to introduce the reader to the concept of comparison algebra, defined as a type of C*-algebra of singular integral operators. The first part of the book develops the necessary elements of the spectral theory of differential operators as well as the basic properties of elliptic second order differential operators. The author then introduces comparison algebras and describes their theory in L2-spaces and L2-Soboler spaces, and in particular their importance in solving functional analytic problems involving differential operators. The book is based on lectures given in Sweden and the USA.

Contents:

Cover; Title; Copyright; Preface; Contents; Chapter 1. Abstract spectral theory in Hilbert spaces; 1.1. Unbounded linear operators on Banach and Hilbert spaces; 1.2. Self-adjoint extensions of hermitian operators; 1.3. On the spectral theorem for self-adjoint operators.; 1.4. Proof of the spectral theorem; 1.5. A result on powers of positive operators; 1.6. On HS-chains 2; Chapter 2. Spectral theory of differential operators; 2.1. Linear differential operators on a subdomain of ln; 2.2. Generalized boundary problems; ordinary differential expressions

2.3. Singular endpoints of a 2r-th order Sturm-Liouville problem2.4. The spectral theorem for a second order expression; Chapter 3. Second order elliptic expressions on manifolds; 3.1. 2-nd order partial differential expressions on manifolds; Weyl!s lemma; Dirichlet operator; 3.2. Boundary regularity for the Dirichlet realization; 3.3. Compactness of the resolvent of the Friedrichs extension; 3.4. A Green's function for H and Hd and a mean value inequality; 3.5. Harnack inequality; Dirichlet problem; maximum principle; 3.6. Change of dependent variable; normal forms

positivity of the Green's functionChapter 4. Essential self-adjointness of the Minimal Operator; 4.1. Essential self-adjointness of powers of H0; 4.2. Essential self-adjointness of HQ; 4.3. Proof of theorem 1.1; 4.4. Proof of Frehse's theorem; 4.5. More criteria for essential self-adjointness; Chapter 5. C -Comparison algebras; 5.1. Comparison operators and comparison algebras; 5.2. Differential expressions of order _< 2; 5.3. Compactness criteria for commutators; 5.4. Comparison algebras with compact commutators; 5.5. A discussion of one-dimensional problems

5.6. An expansion for expressions within reach of an algebra CChapter 6. Minimal comparison algebra and wave front space.; 6.1. The local invariance of the minimal comparison algebra; 6.2. The wave front space; 6.3. Differential expressions within reach of the algebra J0; 6.4. The Sobolev estimate for elliptic expressions expressions on a compact; Chapter 7. The secondary symbol space; 7.1. The symbol space of a general comparison algebra; 7.2. The space flAw , and some examples; 7.3. Stronger conditions and more detail on M\W .; 7.4. More structure of ffi , and more on examples

Chapter 8. Comparison algebras with non-compact commutators8.1. An algebra invariant under a discrete translation group; 8.2. A C -algebra on a poly-cylinder; 8.3. Algebra surgery; 8.4. Complete Riemannian manifolds with cylindrical ends; Chapter 9. H -Algebras; higher order s operators within reach; 9.1. Higher order Sobolev spaces and H -comparison algebras; 9.2. Closer analysis of some of the conditions (1.) and (m.); 9.3. Higher order differential expressions within reach of C or Cs; 9.4. Symbol calculus in W; 9.5. Local properties of the Sobolev spaces Hs

9.6. Sobolev norms of integral order

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographic references and index.

ISBN:

1-139-88601-0

1-107-36601-1

1-107-37074-4

1-107-36110-9

1-107-36925-8

1-299-40381-6

1-107-36355-1

0-511-66283-1