A spectral Schwarz lemma, II

Abstract

The spectral Schwarz lemma holding for any operator-valued holomorphic map of the open unit disc $\Delta \subset \mathbb{C}$ into the algebra of all continuous linear operators acting on a complex Banach space is extended to other scalar- and vector-valued gauges of continuous linear operators, with particular attention to the numerical range and the numerical radius of any continuous linear mapping acting on a complex Banach space.