JournalsrlmVol. 19, No. 4pp. 309–323

A spectral Schwarz lemma

  • Edoardo Vesentini

    Pisa, Italy
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Abstract

The classical Schwarz lemma for any scalar-valued holomorphic function hh mapping the open unit disc ΔC\Delta \subset \mathbb{C} into itself is generalized, replacing hh by a holomorphic map ff of Δ\Delta into a unital associative Banach algebra A{\cal A}, and h(z)| h(z) | by the spectral radius of f(z)f(z) (zΔz \in \Delta). If A=L(H){\cal A} = {\cal L}({\cal H}) and H{\cal H} is a complex Hilbert space, the behaviour of the numerical radius of f(z)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