Eigenvalue Distribution for Nonselfadjoint Toeplitz Matrices

Abstract

In this expository article we discuss the question of the limiting distribution as n → ∞ of the the eigenvalues of n × n Toeplitz matrices, with emphasis on the determination of the limiting set and the limiting measure (if these exist). In the selfadjoint case the limiting set is the interval between the essential infimum and the essential supremum of the symbol, and the limiting measure is the canonical measure induced by the symbol. Theorems of Schmidt-Spitzer and Hirschman, which determine these when the symbol is a Laurent polynomial, are discussed. They are quite different from what they are in the selfadjopint case. A conjecture is presented (with some evidence given) that, nevertheless, the limiting measure is “in general” the one induced by the symbol.

Toeplitz Lecture presented at Tel Aviv University, March, 1993