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.

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

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. Invariant Subspaces and Unicellularity of
Operators of Generalized Integration in Spaces of Analytic
Functionals, Math. Notes, 1977, vol. 22, no. 2, pp. 613-618. DOI:
10.1007/BF01142725.

6. Tkachenko V. A. Operators that Commute with Generalized
Integration in Spaces of Analytic Functionals, Math. Notes, 1979,
vol. 25, no. 2, pp. 141-146. DOI: 10.1007/BF01780970.

7. Krasichkov-Ternovskii I. F. Invariant Subspaces of Analytic
Functions. III. On the Extension of Spectral Synthesis, Math.
USSR-Sbornik, 1972, vol. 17, pp. 327-348. DOI:
10.1070/SM1972v017n03ABEH001508.

8. Karaev M. T. Invariant Subspaces, Cyclic Vectors, Commutant and
Extended Eigenvectors of Some Convolution Operators, Methods Funct.
Anal. Topology, 2005, vol. 11, no. 1, pp. 48-59.

9. Karaev M. T. Duhamel Algebras and Applications, Functional
Analysis and its Applications, 2018, vol. 52, no. 1, pp. 1-8. DOI:
doi.org/10.4213/faa3481.

10. Hormander L. The Analysis of Linear Partial Differential
Operators. Vol. I: Distribution Theory and Fourier Analysis,
Springer, 1983, 391 p.

11. Binderman Z. Functional Shifts Induced by Right Invertible
Operators, Math. Nachr., 1992, vol. 157, pp. 211-224. DOI:
10.1002/mana.19921570117.

12. Dimovski I. N., Hristov V. Z. Commutants of the Pommiez
Operator, Int. J. Math. and Math. Science, 2005, no. 8, pp.
1239-1251.

13. Muskheleshvili N. I. Nekotorye osnovnye zadachi
matematicheskoi teorii uprugosti [Some Basic Problems of the
Mathematical Elasticity Theory], Moscow, GRFML, 1966, 708 p. (in
Russian).

14. Edvards R. E. Functional Analysis. Theory and Applications.
N.Y., Holt, Rinehart and Winston, 1965, 791 p.

15. Titchmarsh E. C. The Zeros of Certain Integral Function, Proc.
London Math. Soc., 1926, vol. 25, pp. 283-302.

16. Mikusinski J. Operational Calculus, N.Y., Pergamon Press,
1959, 495 p.

17. Kalish G. K. A Functional Analysis Proof of Titchmarsh's
Theorem on Convolution, J. Math Anal. Appl., 1962, vol. 5, pp.
176-183.

18. Dimovski I. Convolutional Calculus, London, Kluwer, 1990, 184
p.