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 \({A\otimes B}\) are related by the inequalities \({w(A)w(B)\le w(A\otimes B)\le {\rm min}\{\|A\|w(B), w(A)\|B\|\}}\) . In this paper, we show that (1) if \({w(A\otimes B) = w(A)w(B)}\) , 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 \({w(A\otimes B) = w(A)w(B) = \|A\|w(B)}\) . 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–Foias dilation theory.