Toeplitz matrices with slowly growing pseudospectra

Let T(a) be the infinite Toeplitz matrix with the symbol a and let T-n(a) denote the n x n principal submatrix of T(a). The pseudospectra of T-n(a) are known to converge to the pseudaspectrum of T(a) as n --> infinity provided a is piecewise continuous. Only recently, Mark Embree, Nick Trefethen, and one of the authors observed that this convergence may be spectacularly slow in case a has a jump. The main result of this paper says that such a slow convergence of pseudospectra is generic even within the class of continuous symbols.