Numerical radii for tensor products of matrices

Abstract

For n-by-n and m-by-m complex matrices A and B, it is known that the inequality w(A circle times B) = parallel to A parallel to w(B) holds, where w(center dot) and parallel to center dot parallel to 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 parallel to A parallel to = 1 and w(A circle times B) = w(B), then one of the following two conditions holds: (i) A has a unitary part, and (ii) A is completely nonunitary and the numerical range W(B) of B is a circular disc centered at the origin, (2) if parallel to A parallel to = parallel to A(k)parallel to = 1 for some k, 1 <= k < infinity, then w(A) >= cos(pi/(k + 2)), and, moreover, the equality holds if and only if A is unitarily similar to the direct sum of the (k + 1)-by-(k + 1) Jordan block J(k+1) and a matrix B with w(B) <= cos(pi/(k + 2)), and (3) if B is a nonnegative matrix with its real part (permutationally) irreducible, then w(A circle times B) = parallel to A parallel to w(B), if and only if either p (A) = infinity or n (B) = p (A) < infinity and B is permutationally similar to a block-shift matrix [GRAPHICS] with k = n (B), where p (A) = sup{l >= 1 : parallel to A(l)parallel to = parallel to A parallel to(l)} and n (B) = sup{l >= 1 : B-l not equal 0}.