 
Summary: C
ALGEBRAS AND NUMERICAL LINEAR ALGEBRA
William Arveson
Department of Mathematics
University of California
Berkeley, CA 94720 USA
Abstract. Given a self adjoint operator A on a Hilbert space, suppose that that one
wishes to compute the spectrum of A numerically. In practice, these problems often
arise in such a way that the matrix of A relative to a natural basis is "sparse". For
example, discretized second order differential operators can be represented by doubly
infinite tridiagonal matrices. In these cases it is easy and natural to compute the
eigenvalues of large n × n submatrices of the infinite operator matrix, and to hope
that if n is large enough then the resulting distribution of eigenvalues will give a
good approximation to the spectrum of A. Numerical analysts call this the Galerkin
method.
While this hope is often realized in practice it often fails as well, and it can fail
in spectacular ways. The sequence of eigenvalue distributions may not converge as
n , or they may converge to something that has little to do with the original
operator A. At another level, even the meaning of `convergence' has not been made
precise in general. In this paper we determine the proper general setting in which
