Numerical Radii for Tensor Products of Operators

摘要

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 w(A)w(B) ≤ w(A ? B) ≤ min{||A||w(B),w(A)||B||}. In this paper, we show that (1) if w(A?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 ? 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-Foia? dilation theory.