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DOI: 10.23671/VNC.2018.3.17988

# The Commutant of the Pommiez Operator in a Space of Entire Functions of Exponential Type and Polynomial Growth on the Real Line

Ivanova, O. A. , Melikhov, S. N.
Vladikakazian Mathematical Journal 2018. Vol. 20. Issue 3.
Abstract:
In the space of entire functions of exponential type representing a strong dual to a Frechet space of infinitely differentiable functions on a real interval containing the origin, linear continuous operators commuting with the Pommiez operator are investigated. They are given by a continuous linear functional on this space of entire functions and hence, up to the adjoint of the Fourier--Laplace transform, by an infinite differentiable function on the initial interval. A complete characterization of linear continuous functionals defining isomorphisms by virtue of the indicated correspondence is given. It is proved that isomorphisms are determined by functions that do not vanish at the origin (and only by them). An essential role in proving the corresponding criterion is played by a method exploiting the theory of compact operators in Banach spaces. The class of those functions infinitely differentiable on the considered interval that define the operators from the mentioned commutant close to isomorphisms is distinguished. Such operators have finite-dimensional kernels. For an interval other than a straight real line, we also define the class of operators from the commutant of the Pommiez operator that are not surjective. The adjoint of a continuous linear operator that commutes with Pommiez operators is realized in the space of infinitely differentiable functions as an operator obtained by fixing one factor in the Duhamel product. The essential difference of the situation under consideration from the previously studied one is the absence of cyclic vectors of the Pommiez operator in the considered space of entire functions.
Keywords: Pommiez operator, entire function of exponential type, space of infinitely differentiable functions, commutant, isomorphism.