Contractivity of M\"obius functions of operators

TL;DR

This paper characterizes when certain Möbius transformations of operators are contractions, using the numerical range of the inverse operator, with specific results for the Volterra operator and its powers.

Contribution

It provides a new characterization of contraction conditions for Möbius functions of operators based on the numerical range, including special cases for the Volterra operator.

Findings

Characterization of contraction conditions in terms of numerical range.

Specific results for the Volterra operator on L^2[0,1].

Demonstration that powers of the Volterra operator do not produce contractions under these transformations.

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 $\lambda\ne\mu$.