eBook ISBN:  9781470403751 
Product Code:  MEMO/163/777.E 
List Price:  $60.00 
MAA Member Price:  $54.00 
AMS Member Price:  $36.00 
eBook ISBN:  9781470403751 
Product Code:  MEMO/163/777.E 
List Price:  $60.00 
MAA Member Price:  $54.00 
AMS Member Price:  $36.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 163; 2003; 92 ppMSC: Primary 58; 47; 46
The \(0\)calculus on a manifold with boundary is a microlocalization of the Lie algebra of vector fields that vanish at the boundary. It has been used by Mazzeo, Melrose to study the Laplacian of a conformally compact metric. We give a complete characterization of those \(0\)pseudodifferential operators that are Fredholm between appropriate weighted Sobolev spaces, and describe \(C^{*}\)algebras that are generated by \(0\)pseudodifferential operators. An important step is understanding the socalled reduced normal operator, or, almost equivalently, the infinite dimensional irreducible representations of \(0\)pseudodifferential operators. Since the \(0\)calculus itself is not closed under holomorphic functional calculus, we construct submultiplicative Fréchet \(*\)algebras that contain and share many properties with the \(0\)calculus, and are stable under holomorphic functional calculus (\(\Psi^{*}\)algebras in the sense of Gramsch). There are relations to elliptic boundary value problems.
ReadershipGraduate students and research mathematicians interested in analysis.

Table of Contents

Chapters

Part 1. Fredholm theory for $0$pseudodifferential operators

1. Review on basic objects of 0geometry

2. The small 0calculus and the 0calculus with bounds

3. The $b$$c$calculus on an interval

4. The reduced normal operator

5. Weighted 0Sobolev spaces

6. Fredholm theory for 0pseudodifferential operators

Part 2. Algebras of $0$pseudodifferential operators of order $0$

7. $C$*algebras of 0pseudodifferential operators

8. $\Psi $*algebras of 0pseudodifferential operators


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The \(0\)calculus on a manifold with boundary is a microlocalization of the Lie algebra of vector fields that vanish at the boundary. It has been used by Mazzeo, Melrose to study the Laplacian of a conformally compact metric. We give a complete characterization of those \(0\)pseudodifferential operators that are Fredholm between appropriate weighted Sobolev spaces, and describe \(C^{*}\)algebras that are generated by \(0\)pseudodifferential operators. An important step is understanding the socalled reduced normal operator, or, almost equivalently, the infinite dimensional irreducible representations of \(0\)pseudodifferential operators. Since the \(0\)calculus itself is not closed under holomorphic functional calculus, we construct submultiplicative Fréchet \(*\)algebras that contain and share many properties with the \(0\)calculus, and are stable under holomorphic functional calculus (\(\Psi^{*}\)algebras in the sense of Gramsch). There are relations to elliptic boundary value problems.
Graduate students and research mathematicians interested in analysis.

Chapters

Part 1. Fredholm theory for $0$pseudodifferential operators

1. Review on basic objects of 0geometry

2. The small 0calculus and the 0calculus with bounds

3. The $b$$c$calculus on an interval

4. The reduced normal operator

5. Weighted 0Sobolev spaces

6. Fredholm theory for 0pseudodifferential operators

Part 2. Algebras of $0$pseudodifferential operators of order $0$

7. $C$*algebras of 0pseudodifferential operators

8. $\Psi $*algebras of 0pseudodifferential operators