A spectral Schwarz lemma

Abstract

The classical Schwarz lemma for any scalar-valued holomorphic function $h$ mapping the open unit disc $\Delta \subset \mathbb{C}$ into itself is generalized, replacing $h$ by a holomorphic map $f$ of $\Delta$ into a unital associative Banach algebra ${\cal A}$, and $| h(z) |$ by the spectral radius of $f(z)$ ($z \in \Delta$). If ${\cal A} = {\cal L}({\cal H})$ and ${\cal H}$ is a complex Hilbert space, the behaviour of the numerical radius of $f(z)$ is also investigated.