# A spectral Schwarz lemma

### Edoardo Vesentini

Pisa, Italy

## 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.

## Cite this article

Edoardo Vesentini, A spectral Schwarz lemma. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 19 (2008), no. 4, pp. 309–323

DOI 10.4171/RLM/527