Numerical Radii for Tensor Products of Operators

Abstract

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 are related by the inequalities . In this paper, we show that (1) if , then w(A) = rho(A) or w(B) = rho(B), where rho(center dot) denotes the spectral radius of an operator, and (2) if A is hyponormal, then . 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-FoiaAY dilation theory.