Eigenvalues and pseudo-eigenvalues of Toeplitz matrices (Q1183184)

The \(\varepsilon\)-pseudo-eigenvalues of an \(N\times N\) matrix \(A\) are those complex numbers \(z\) for which \(\|(zI-A)^{-1}\|_ 2\geq 1/\varepsilon>0\). The authors' results, which are partly empirical, show that, for small \(\varepsilon\) and large \(N\), the \(\varepsilon\)- pseudospectrum of a Toeplitz matrix is roughly the same as the spectrum of the associated Toeplitz operator. The corresponding pseudo- eigenvectors are also investigated. The authors argue that the existing very different results on the exact spectra of nonnormal Toeplitz matrices are of dubious practical significance.