##### Department of Mathematics,

University of California San Diego

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### Analysis Colloquium

## Louis Boutet de Monvel

#### University of Paris

## Logarithmic trace of the Szego and Toeplitz projectors

##### Abstract:

In my book with V. Guillemin ``The Spectral Theory of Toeplitz Operators'', we defined Toeplitz projectors on a compact contact manifold, which are analogues of the Szego projector on a strictly pseudo-convex boundary. If $X$ is a contact manifold, the kernel of a Toeplitz projector, just as the Szego kernel, has a holonomic singularity including a polar and a logarithmic term, and its trace is well defined. \vskip .1in \noindent We show that the trace of a Toeplitz operator only depends on the contact structure of $X$. If $X$ is the three sphere equipped with any contact form, this invariant vanishes (Y. Eliashberg has shown there are many nonisomorphic such contact forms); this makes it not unlikely that the trace vanishes identically.

Host: Salah Baouendi

### April 26, 2005

### 10:30 AM

### AP&M 6438

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