Curvature in Multivariable Operator Theory
by
William Arveson
University of California, Berkeley

The Gauss Bonnet formula asserts that the integral of the Gaussian curvature over a compact oriented Riemann surface is an integer, namely the alternating sum of the three Betti numbers of the surface...the so-called Euler characteristic. This remarkable theorem was generalized in the late forties to higher dimensional manifolds, most notably by S.-S. Chern.

After reviewing these geometric issues for analysts, we discuss some basic constructions with commuting sets of operators acting on a common Hilbert space H. Such an H is a module over the algebra of complex polynomials in the corresponding number of variables, and is called a Hilbert module. When it is of ``finite rank" in an appropriate sense it is possible to define a numerical invariant K(H) which is analogous to the average curvature of a manifold. This invariant is a nonnegative real number, it has some interesting properties, it is not easy to compute, and it has a tendency to be an integer.

We also introduce an operator-theoretic analogue of the Euler characteristic of a finite rank Hilbert module H. The Euler characteristic depends only on the linear algebra of a certain complex vector space acted upon by a commuting set of linear transformations...and which is finitely generated in the sense of commutative algebra. The Euler characteristic is an integer and there is well-established technology for computing it (going back to Hilbert's work of the 1890s). The central result is that for ``graded" Hilbert modules H, K(H) agrees with the Euler characteristic of H. The proof is based on asymptotic formulas for K(H) and the Euler characteristic.

Date received: May 10, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacw-69.