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 $$��(AB)\le r(A)r(B) \quad\text{ for all bounded linear operators } B$$ if and only if there is a unique $��\in ��(A)$ satisfying $|��| = ��(A)$ and $A = \frac{��(I + L)}{2}$ for a contraction $L$ with $1\in��(L)$. One can get the same conclusion on $A$ if $��(AB) \le r(A)r(B)$ 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 $x$.

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