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 $\mathcal{A}$, and $|h(z)|$ by the spectral radius of $f(z)$ ($z\in \Delta$). If $\mathcal{A}=\mathcal{L}(\mathcal{H})$ and $\mathcal{H}$ is a complex Hilbert space, the behaviour of the numerical radius of $f(z)$ is also investigated.