Polynomial bounds for small matrices

Abstract

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.

Keywords:

• polynomially bounded matrices
• completely bounded norm

AMS Subject Classifications:

• 46L07
• 47L25

Acknowledgments

For various forms of support during the course of this work, the authors wish to thank the following: NSERC of Canada, the Maui High Performance Computing Center (MHPCC), the University of Guelph sabbatical programme and the Indian Statistical Institute (Delhi Centre). The July 2012 WONRA meeting, held in Kaohsiung, Taiwan, was also stimulating; in particular the relation between and played a role in Anne Greenbaum’s lecture about the Crouzeix conjecture. This paper is dedicated to Pei YuanWu on the occasion of his 65th birthday.