The spectral Schwarz lemma holding for any operator-valued holomorphic map of the open unit disc 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.
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Edoardo Vesentini, A spectral Schwarz lemma, II. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 23 (2012), no. 1, pp. 89–95DOI 10.4171/RLM/618