Spectral radius, numerical radius, and the product of operators

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 $$rho(AB)le r(A)r(B) quadtext{ for all bounded linear operators } B$$ 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 $1insigma(L)$. 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 $x$.