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On the Spectral Value of Semigroups of Holomorphic Functions

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Abstract

Let \((\phi _t)_{t \ge 0}\) be a semigroup of holomorphic self-maps of the unit disk \({{\,\mathrm{{\mathbb {D}}}\,}}\) with Denjoy–Wolff point \(\tau =1\). The angular derivative is \(\phi _t^{\prime }(1)= e^{-\lambda t}\), where \(\lambda \ge 0\) is the spectral value of \((\phi _t)\). If \(\lambda >0\) the semigroup is hyperbolic, otherwise it is parabolic. Suppose K is a compact non-polar subset of \({{\,\mathrm{{\mathbb {D}}}\,}}\). We specify the type of the semigroup by examining the asymptotic behavior of \(\phi _t(K)\). We provide a representation of the spectral value of the semigroup with the use of several potential theoretic quantities, e.g., harmonic measure, Green function, extremal length, condenser capacity.

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Acknowledgements

The author wishes to thank the anonymous referee for the valuable insight and constructive comments that improved this paper. Moreover, the author would like to thank the Department of Mathematics, University of Wuerzburg, where part of this research was carried out. This research is co-financed by Greece and the European Union (European Social Fund—ESF) through the Operational Programme “Human Resources Development, Education and Lifelong Learning” in the context of the project “Reinforcement of Post-doctoral researchers—2nd Cycle” (MIS-5033021), implemented by the State Scholarships Foundation (IKY).

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  1. Department of Mathematics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece

    Maria Kourou

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  1. Maria Kourou
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Correspondence to Maria Kourou.

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Kourou, M. On the Spectral Value of Semigroups of Holomorphic Functions. J Geom Anal 31, 10473–10497 (2021). https://doi.org/10.1007/s12220-021-00653-w

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