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 $$\rho(AB)\le r(A)r(B) \quad\text{ for all bounded linear operators } B$$ 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 $x$.