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

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	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. Vladikavkaz 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. Language: Russian Download the full text For citation: Ivanova O. A., Melikhov S. N. The Commutant of the Pommiez Operator in a Space of Entire Functions of Exponential Type and Polynomial Growth on the Real Line, Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 20, no. 2, pp.48-56. DOI 10.23671/VNC.2018.3.17988 + References 1. Ivanova O. A., Melikhov S. N. On Operators Commuting with a Pommiez type Operator in Weighted Spaces of Entire Functions, St. Petersburg Math. J., 2017, vol. 28, no. 2, pp. 209-224. DOI: 10.1090/spmj/1447. 2. Ivanova O. A., Melikhov S. N. On an Algebra of Analytic Functionals Connected with a Pommiez Operator, Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], 2016, vol. 18, no. 4, pp. 34-40 (in Russian). DOI 10.23671/VNC.2016.4.5989. 3. Ivanova O. A., Melikhov S. N. On Invarint Subspaces of the Pommiez Operator in Spaces of Entire Functions of Exponential Type, Complex Analysis, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., vol. 142, Moscow, VINITI, 2017, pp. 111-120 (in Russian). 4. Ivanova O. A., Melikhov S. N. On the Completeness of Orbits of a Pommiez Operator in Weighted (LF)-Spaces of Entire Functions, Complex Analysis and Operator Theory, 2017, vol. 11, pp. 1407-1424. DOI: 10.1007/s11785-016-0617-5. 5. Tkachenko V. A. 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Convolutional Calculus, London, Kluwer, 1990, 184 p. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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