Polynomial bounds for small matrices
Michel Crouzeix, Frank L. Gilfeather, John Holbrook · Linear and Multilinear Algebra · 2013
The polynomial bound of a Hilbert space operator is the quantity . One of the Halmos ‘Ten problems’ asked whether implies that is similar to a contraction. This question was settled in the negative by Pisier after many years. The finite–dimensional version of the problem asks to what extent can be exceeded by . We establish some general criteria for the equality and show that can occur even for matrices.