 

Kent E. Morrison
Spectral Approximation of Multiplication Operators


Published: 
March 20, 1995 
Keywords: 
Eigenvalues, spectrum, multiplication operators, Toeplitz matrices, Walsh functions 
Subject: 
Primary: 4702; Secondary: 15A60, 4704, 47B35, 47B38, 65F15 


Abstract
A linear operator on a Hilbert space may be approximated with finite
matrices by choosing an orthonormal basis of the Hilbert space. For an
operator that is not compact such approximations cannot converge in the
norm topology on the space of operators. Multiplication operators on spaces
of L_{2} functions are never compact; for them we consider how well the
eigenvalues of the matrices approximate the spectrum of the multiplication
operator, which is the essential range of the multiplier. The choice of the
orthonormal basis strongly affects the convergence. Toeplitz matrices arise
when using the Fourier basis of exponentials exp(ikθ). We also
consider the basis of Legendre polynomials and the basis of Walsh
functions.


Author information
Department of Mathematics, California Polytechnic State University, San Luis Obispo, CA 93407
kmorriso@oboe.calpoly.edu

