Numerical Radii for Tensor Products of Operators

For bounded linear operators A and B on Hilbert spaces H and K, respectively, it is known that the numerical radii of A, B and A ⊗ B are related by the inequalities (Formula presented.). In this paper, we show that (1) if (Formula presented.), then w(A) = ρ(A) or w(B) = ρ(B), where ρ(·) denotes the spectral radius of an operator, and (2) if A is hyponormal, then (Formula presented.). Here (2) confirms a conjecture of Shiu's and is proven via dilating the hyponormal A to a normal operator N with the spectrum of N contained in that of A. The latter is obtained from the Sz.-Nagy-Foiaş dilation theory.