Matthias Lesch

On the Noncommutative Residue for Pseudodifferential Operators with log-Polyhomogeneous Symbols

AbstractWe study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion

\eta

a \sim \sum\limits_{j = 0}^\infty {a_{m - j,} } a_{m - j,} (x,\xi ) = \sum\limits_{l = 0}^k {a_{m - j,l} (x,\xi )} \log ^l |\xi |,

On the Determinant of One-Dimensional Elliptic Boundary Value Problems

where am-j,l is homogeneous in ξ of degree m-j. We call these symbols log-polyhomogeneous. We will explain why this algebra of pseudodifferential operators is natural.We study log-polyhomogeneous functions on symplectic cones and generalize the symplectic residue of Guillemin to these functions. Similarly, as for homogeneous functions, for a log-polyhomogeneous function, this symplectic residue is an obstruction against being a sum of Poisson brackets.For a pseudodifferential operator with log-polyhomogeneous symbol, A, and a classical elliptic pseudodifferential operator, P, we show that the generalized ζ-function Tr(AP-s) has a meromorphic continuation to the whole complex plane, however possibly with higher-order poles.Our algebra of operators has a bigrading given by the order and the highest log-power occuring in the symbol expansion. We construct “higher” noncommutative residue functionals on the subspaces given by the log-grading. However, in contrast to the classical case we prove that the whole algebra does not admit any nontrivial traces.Finally, we show that an analogue of the Kontsevich–Vishik trace also exists for our algebra. Our method also provides an alternative approach to the Kontsevich–Vishik trace.

A local global principle for regular operators in Hilbert C⁎-modules

Motivated by the work of Vishik on the analytic torsion we introduce a new class of generalized Atiyah-Patodi-Singer boundary value problems. We are able to derive a full heat expansion for this class of operators generalizing earlier work of Grubb and Seeley. As an application we give another proof of the gluing formula for the eta invariant. Our class of boundary conditions contains as special cases the usual (nonlocal) Atiyah-Patodi-Singer boundary value problems as well as the (local) relative and absolute boundary conditions for the Gauss-Bonnet operator.

The Inverse Spectral Problem for First Order Systems on the Half Line

Abstract:We discuss the ζ-regularized determinant of elliptic boundary value problems on a line segment. Our framework is applicable for separated and non-separated boundary conditions.

The Calderón projection: New definition and applications

Abstract Hilbert C ⁎ -modules are the analogues of Hilbert spaces where a C ⁎ -algebra plays the role of the scalar field. With the advent of Kasparovʼs celebrated KK -theory they became a standard tool in the theory of operator algebras. While the elementary properties of Hilbert C ⁎ -modules can be derived basically in parallel to Hilbert space theory the lack of an analogue of the Projection Theorem soon leads to serious obstructions and difficulties. In particular the theory of unbounded operators is notoriously more complicated due to the additional axiom of regularity which is not easy to check. In this paper we present a new criterion for regularity in terms of the Hilbert space localizations of an unbounded operator. We discuss several examples which show that the criterion can easily be checked and that it leads to nontrivial regularity results.

SPECTRAL FLOW AND THE UNBOUNDED KASPAROV PRODUCT

On the half line [0, ∞) we study first order differential operators of the form