Contractivity of Möbius functions of operators

Let T be an injective bounded linear operator on a complex Hilbert space. We characterize the complex numbers $\lambda ,\mu$ for which $(I+\lambda T){(I+\mu T)}^{-1}$ is a contraction, the characterization being expressed in terms of the numerical range of the possibly unbounded operator $T^{-1}$.

When $T=V$, the Volterra operator on $L^2[0,1]$, this leads to a result of Khadkhuu, Zemánek and the second author, characterizing those $\lambda ,\mu$ for which $(I+\lambda V){(I+\mu V)}^{-1}$ is a contraction. Taking $T=V^n$, we further deduce that $(I+\lambda V^n){(I+\mu V^n)}^{-1}$ is never a contraction if $n\ge 2$ and $\lambda \ne \mu$.