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Phone: (407) 823-6284;   Fax: (407) 823-6253;   MAP  207 2002-2003 MATHEMATICS COLLOQUIUM SERIES 12/06/02 Colloquium DR. B.V. LIMAYE Indian Institute of Technology Bombay India Spectral Approximation for Compact Integral Operators Abstract:   Several questions in science and engineering can be modeled in terms of a function space X and a compact integral operator T on it. The Spectral Problem for such an operator T consists of finding its nonzero eigenvalues as well as bases for the corresponding spectral subspaces. A compact integral operator T can often be approximated by a sequence of continuous operators , all defined on X and having finite ranks. A newly developed mode of convergence of to T allows one to approximate nonzero eigenvalues of T as well as bases for the corresponding spectral subspaces. A canonical procedure for reducing the Spectral Problem for a continuous operator having finite rank to the Spectral Problem for a matrix is outlined. Also, two methods for improving the accuracy of the approximation of the nonzero eigenvalues T are indicated. A couple of numerical examples illustrate some of these ideas.