Abstract
We determine the spectrum of the Voltterra-type integral operators \(V_g\) on the growth type Fock–Sobolev spaces \(\mathcal {F}_{\psi _m}^\infty \). We also characterized the bounded and compact spectral properties of the operators in terms of function-theoretic properties of the inducing map g. As a means to prove our main results, we first described the spaces in terms of Littlewood–Paley type formula which is interest of its own.
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Notes
The notation \(U(z)\lesssim V(z)\) (or equivalently \(V(z)\gtrsim U(z)\)) means that there is a constant C such that \(U(z)\le CV(z)\) holds for all z in the set of a question. We write \(U(z)\simeq V(z)\) if both \(U(z)\lesssim V(z)\) and \(V(z)\lesssim U(z)\).
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Communicated by Jussi Behrndt.
The author is partially supported by HSH Grant 1244/H15.
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Mengestie, T. On the Spectrum of Volterra-Type Integral Operators on Fock–Sobolev Spaces. Complex Anal. Oper. Theory 11, 1451–1461 (2017). https://doi.org/10.1007/s11785-016-0629-1
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DOI: https://doi.org/10.1007/s11785-016-0629-1