A note on properties and computations of matrix pseudospectra

Recently, Shen et al. has published a paper concerning the properties and computations of matrix pseudospectra. However, some of the main results seem to be incomplete. As a complement of the paper, in this note we first make several examples to show the incompleteness of that results, then we give the modified results through giving more appropriate definitions replacing the one given in [Y. Shen, J. Zhao, H. Fan, Properties and computations of matrix pseudospectra, Applied Mathematics and Computation 161 (2005) 385–393]. An interesting inclusion theorem for pseudospectra is given, which provides preferable bounds for standard pseudospectra. Some other properties are also explored.