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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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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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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DOI: https://doi.org/10.1007/s11785-018-0820-7
Keywords
- Weighted Fock space
- Generalized Fock spaces
- Volterra operator
- Multiplication operator
- Differential operator
- Bounded
- Compact