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Integral, Differential and Multiplication Operators on Generalized Fock Spaces

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Abstract

Volterra companion integral and multiplication operators with holomorphic symbols are studied for a large class of generalized Fock spaces on the complex plane \(\mathbb {C}\). The weights defining these spaces are radial and subject to a mild smoothness condition. In addition, we assumed that the weights decay faster than the classical Gaussian weight. One of our main results show that there exists no nontrivial holomorphic symbols g which induce bounded Volterra companion integral \(I_g\) and multiplication operators \(M_g\) acting between the weighted spaces. We also describe the bounded and compact Volterra-type integral operators \(V_g\) acting between \({\mathcal {F}}_q^\psi \) and \({\mathcal {F}}_p^\psi \) when at least one of the exponents p or q is infinite, and extend results of Constantin and Peláez for finite exponent cases. Furthermore, we showed that the differential operator D acts in unbounded fashion on these and the classical Fock spaces.

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Acknowledgements

We would like to thank the referee for careful review of our paper and pointing us relevant literatures, which eventually helped us put our work in context to already known results.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Western Norway University of Applied Sciences, Klingenbergvegen 8, 5414, Stord, Norway

    Tesfa Mengestie

  2. Toki University, Hitachi, 316-8511, Japan

    Sei-Ichiro Ueki

Authors
  1. Tesfa Mengestie
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  2. Sei-Ichiro Ueki
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Corresponding author

Correspondence to Tesfa Mengestie.

Additional information

Communicated by Daniel Aron Alpay.

The first author was supported by HSH Grant 1244/H15, and the second author’s work was partially supported by JSPS KAKENHI Grant 26800050.

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Mengestie, T., Ueki, SI. Integral, Differential and Multiplication Operators on Generalized Fock Spaces. Complex Anal. Oper. Theory 13, 935–958 (2019). https://doi.org/10.1007/s11785-018-0820-7

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