C*-algebras and pseudospectra of large Toeplitz matrices

In contrast to spectra, pseudospectra of large Toeplitz matrices

behave as nicely as one could ever expect. We demonstrate some

basic phenomena of the asymptotic distribution of the spectra

and pseudospectra of Toeplitz matrices and show how by employing

a few simple $C^*$-algebra arguments one can prove rigorous

convergence results for the pseudospectra. The talk is a survey

of the development since a 1992 paper by Reichel and Trefethen

and is not addressed to specialists, but rather to a general

mathematically interested audience.