Numerical radii for tensor products of matrices

Abstract

For-by- and-by- complex matrices and, it is known that the inequality holds, where and denote, respectively, the numerical radius and the operator norm of a matrix. In this paper, we consider when this becomes an equality. We show that (1) if and, then one of the following two conditions holds: (i) has a unitary part, and (ii) is completely nonunitary and the numerical range of is a circular disc centered at the origin, (2) if for some , , then , and, moreover, the equality holds if and only if is unitarily similar to the direct sum of the -by- Jordan block and a matrix with , and (3) if is a nonnegative matrix with its real part (permutationally) irreducible, then , if and only if either or and is permutationally similar to a block-shift matrixwith , where and .

Keywords:

• numerical range
• numerical radius
• tensor product
• -matrix
• nonnegative matrix

AMS Subject Classifications:

• 15A60
• 15A69
• 15B48

Acknowledgments

This paper was presented by the third author at the fourth International Conference on Matrix Analysis and applications in Konya, Turkey. He thanks the organizers for their works with the conference.

Added in proof

The first author has since found a simpler proof of Theorem 2.10 for any bounded linear operator on a Hilbert space with and for some and some unit vector in .

Notes

This research was partially supported by the National Science Council of the Republic of China [grant numbers NSC-101-2115-M-008-006, NSC-101-2115-M-009-001 and NSC-101-2115-M-009- 004]. The third author was also supported by the MOE-ATU.