Spectral radius, numerical radius, and the product of operators

Abstract

Let $\sigma (A)$, $\rho (A)$ 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 $$\rho (AB)\le r(A)r(B)\text{ for all bounded linear operators }B$$ if and only if there is a unique $\mu \in \sigma (A)$ satisfying $|\mu |=\rho (A)$ and $A=\frac{\mu (I+L)}{2}$ for a contraction $L$ with $1\in \sigma (L)$. One can get the same conclusion on $A$ if $\rho (AB)\le r(A)r(B)$ for all rank one operators $B$. If $H$ is of finite dimension, we can further decompose $L$ as a direct sum of $C\oplus 0$ under a suitable choice of orthonormal basis so that $Re(C^{-1}x,x)\ge 1$ for all unit vector $x$.

9 pages