Spectral properties of banded Toeplitz matrices * By Albrecht Bottcher and Sergei M. Grudsky: * 411 pp., US$ 95.00, ISBN 0-89871-599-7 * (Society for Industrial and Applied Mathematics, Philadelphia, 2005)

Recall that an infinite Toeplitz matrix, , say, is one for which the values t_jk dependonly on j – k, so we may write t_jk = a_{j – k}; thusthe entries are constant on diagonals sloping down from topleft to bottom right:

Sucha matrix is said to be banded if there are only finite manynon-zero a_j, say . The propertiesof such a matrix, when defining a linear operator T on a sequencespace, are often closely related to the properties of the Laurentpolynomial

for zlying on the unit circle in the complex plane. For example,suppose that the matrix acts on the most familiar sequence space,the Hilbert space ²; then it has been known for many years thatthe operator norm of T is the maximum