A spectral Schwarz lemma (Q993413)

Summary: 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.