Contractivity of M\"obius functions of operators

Abstract

Let $T$ be a 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, Zem\'anek 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 $n\ge2$ and $\lambda\ne\mu$.