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Strongly Mixing Convolution Operators on Fréchet Spaces of Holomorphic Functions

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Abstract

A theorem of Godefroy and Shapiro states that non-trivial convolution operators on the space of entire functions on \({\mathbb{C}^n}\) are hypercyclic. Moreover, it was shown by Bonilla and Grosse-Erdmann that they have frequently hypercyclic functions of exponential growth. On the other hand, in the infinite dimensional setting, the Godefroy–Shapiro theorem has been extended to several spaces of entire functions defined on Banach spaces. We prove that on all these spaces, non-trivial convolution operators are strongly mixing with respect to a gaussian probability measure of full support. For the proof we combine the results previously mentioned and we use techniques recently developed by Bayart and Matheron. We also obtain the existence of frequently hypercyclic entire functions of exponential growth.

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Authors and Affiliations

  1. Departamento de Matemática-Pab I, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires, 1428, Ciudad Autónoma de Buenos Aires, Argentina

    Santiago Muro & Martín Savransky

  2. CONICET, Buenos Aires, Argentina

    Santiago Muro, Damián Pinasco & Martín Savransky

  3. Departamento de Matemáticas y Estadística, UniversidadTorcuato di Tella, Av. F. Alcorta 7350, 1428, Ciudad Autónomade Buenos Aires, Argentina

    Damián Pinasco

Authors
  1. Santiago Muro
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  2. Damián Pinasco
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  3. Martín Savransky
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Correspondence to Martín Savransky.

Additional information

Partially supported by PIP 2010-2012 GI 11220090100624, PICT 2011-1456, UBACyT 20020100100746, ANPCyT PICT 11-0738 and CONICET.

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Muro, S., Pinasco, D. & Savransky, M. Strongly Mixing Convolution Operators on Fréchet Spaces of Holomorphic Functions. Integr. Equ. Oper. Theory 80, 453–468 (2014). https://doi.org/10.1007/s00020-014-2182-5

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  • DOI: https://doi.org/10.1007/s00020-014-2182-5

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