Numerical radii for tensor products of matrices

Abstract: For (Formula presented.) -by- (Formula presented.) and (Formula presented.) -by- (Formula presented.) complex matrices (Formula presented.) and (Formula presented.) , it is known that the inequality (Formula presented.) holds, where (Formula presented.) and (Formula presented.) 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 (Formula presented.) and (Formula presented.) , then one of the following two conditions holds: (i) (Formula presented.) has a unitary part, and (ii) (Formula presented.) is completely nonunitary and the numerical range (Formula presented.) of (Formula presented.) is a circular disc centered at the origin, (2) if (Formula presented.) for some (Formula presented.) , (Formula presented.) , then (Formula presented.) , and, moreover, the equality holds if and only if (Formula presented.) is unitarily similar to the direct sum of the (Formula presented.) -by- (Formula presented.) Jordan block (Formula presented.) and a matrix (Formula presented.) with (Formula presented.) , and (3) if (Formula presented.) is a nonnegative matrix with its real part (permutationally) irreducible, then (Formula presented.) , if and only if either (Formula presented.) or (Formula presented.) and (Formula presented.) is permutationally similar to a block-shift matrix (Formula presented.) with (Formula presented.) , where (Formula presented.) and (Formula presented.).