Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4615472 | Journal of Mathematical Analysis and Applications | 2015 | 7 Pages |
Abstract
Let Ï(A), Ï(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 satisfiesÏ(AB)â¤r(A)r(B)for all bounded linear operators B if and only if there is a unique μâÏ(A) satisfying |μ|=Ï(A) and A=μ(I+L)2 for a contraction L with 1âÏ(L). One can get the same conclusion on A if Ï(AB)â¤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 Câ0 under a suitable choice of orthonormal basis so that Re(Câ1x,x)â¥1 for all unit vector x.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Rahim Alizadeh, Mohammad B. Asadi, Che-Man Cheng, Wanli Hong, Chi-Kwong Li,