Spectral radius, numerical radius, and the product of operators

Let σ(A), ρ(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 satisfiesρ(AB)≤r(A)r(B)for all bounded linear operators B if and only if there is a unique μ∈σ(A) satisfying |μ|=ρ(A) and A=μ(I+L)2 for a contraction L with 1∈σ(L). One can get the same conclusion on A if ρ(AB)≤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⊕0 under a suitable choice of orthonormal basis so that Re(C−1x,x)≥1 for all unit vector x.