A spectral Schwarz lemma. II (Q2428061)

In Part I [the author, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 19, No. 4, 309--323 (2008; Zbl 1194.30041)], a spectral Schwarz lemma was established for holomorphic operator-valued functions. It was based on the fact that the spectral radius \(\rho(X)\) of a continuous linear operator \(X\) on a complex Banach space is a plurisubharmonic function. In the present paper, this is extended to other gauges associated to \(X\), with particular attention to the numerical range and numerical radius.