SPECTRAL RADIUS AND NUMERICAL RADIUS OF OPERATORS PRODUCT
Rahim Alizadeh, Mohammad B. Asadi, Che‐Man Cheng, Wanli Hong, Chi-Kwong Li · arXiv (Cornell University) · 2014
Let �(A), �(A) and r(A) denote the spectrum, spectral radius and numerical radius of a bounded linear operator A on a Hilbert space H, respectively. We show that a linear operator A satisfying �(AB) � r(A)r(B) for all bounded linear operators B if and only if there is a unique µ 2 �(A) satisfying |µ| = �(A) and A = µ(I+L) 2 for a contraction L with 1 2 �(L). One can get the same conclusion on A if �(AB) � r(A)r(B) for all rank one operators B. If H is of finite dimension, we can further decompose L as a direct sum of C�0 under a suitable choice of orthonormal basis so that Re(C 1 x,x) � 1 for all unit vector x.