Result: asymptotic spectra of dense toeplitz matrices are unstable: Asymptotic spectra of dense Toeplitz matrices are unstable

An infinite Toeplitz matrix is defined by \(T(a)=(a_{j-k}) _{j,k=0}^{\infty}\) where \(\{ a_{k} \} _{k\in \mathbb{Z}}\) are the Fourier coefficients of \(a(e^{i\theta}).\) It is well known that its spectrum is the range of its symbol \(a(\mathbf{T})\). The authors study the approximation of the spectra obtained from the finite section, which is to examine the convergence of the set of eigenvalues of the finite matrix \(T_{n}(a)=(a_{j-k}) _{j,k=0}^{n-1}\). To this end they consider the asymptotic spectrum of \(\Lambda \left( a\right) =\lim \sup_{n\rightarrow \infty }\sigma (T_{n}(a)) \) and show that \(\Lambda \) is discontinuous on the space of continuous symbols. A numerical example is used to illustrate that fact.