Journal of Spectral Theory
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Eigenvalues of Hermitian Toeplitz matrices with polynomially increasing entries
Johan M. Bogoya (1), Albrecht Böttcher (2) and Sergei M. Grudsky (3)
(1) Departamento de Matemáticas, Pontificia Universidad Javeriana, BOGOTA, COLOMBIA(2) Fakultät für Mathematik, Technische Universität Chemnitz, Reichenhainer Strasse 39, D-09107, CHEMNITZ, GERMANY
(3) Departamento de Matemáticas, CINVESTAV del IPN, Apartado Postal 14-740, 07000, MEXICO, D.F., MEXICO
The paper is concerned with finite Hermitian Toeplitz matrices whose entries in the first row grow like a polynomial. Such matrices cannot be viewed as truncations of an infinite Toeplitz matrix which is generated by an integrable function or a nice measure. The main results describe the first-order asymptotics of the extreme eigenvalues as the matrix dimension goes to infinity and also deliver unexpected barriers for the eigenvalues. One purpose of the paper is to popularize once more that questions on the eigenvalues of matrices can be answered in a very elegant way by passing to integral operators. This idea was introduced by Harold Widom about fifty years ago. In this way one can also give an alternative proof to results by William F. Trench on Hermitian Toeplitz matrices with increasing entries.
Keywords: Toeplitz matrix, integral operator, circulant matrix, eigenvalue