Memoirs of the American Mathematical Society 2003; 92 pp; softcover Volume: 163 ISBN10: 0821832727 ISBN13: 9780821832721 List Price: US$60 Individual Members: US$36 Institutional Members: US$48 Order Code: MEMO/163/777
 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. Readership Graduate students and research mathematicians interested in analysis. Table of Contents Part 1. Fredholm theory for \(0\)pseudodifferential operators  Review of basic objects of \(0\)geometry
 The small \(0\)calculus and the \(0\)calculus with bounds
 The \(b\)\(c\)calculus on an interval
 The reduced normal operator
 Weighted \(0\)Sobolev spaces
 Fredholm theory for \(0\)pseudodifferential operators
Part 2. Algebras of \(0\)pseudodifferential operators of order \(0\)  \(C^*\)algebras of \(0\)pseudodifferential operators
 \(\Psi^*\)algebras of \(0\)pseudodifferential operators
 Appendix A. Spaces of conormal functions
 Bibliography
 Notations
 Index
