×

Invariant subspaces in half-plane. (Russian. English summary) Zbl 1463.30118

Ufim. Mat. Zh. 12, No. 3, 30-44 (2020); translation in Ufa Math. J. 12, No. 3, 30-43 (2020).
Summary: We study subspaces of functions analytic in a half-plane and invariant with respect to the differentiation operator. A particular case of an invariant subspace is a space of solutions a linear homogeneous differential equation with constant coefficients. It is known that each solution of such equations is a linear combination of primitive solutions, which are exponential monomials with exponents being possibly multiple zeroes of a characteristic polynomial. The existence of such representation is called Euler fundamental principle. Other particular cases of invariant subspaces are spaces of solutions of linear homogeneous differential, difference and differential-difference equations with constant coefficients of both finite and infinite orders as well as of more general convolution equations and the systems of them. In the work we study the issue on fundamental principle for arbitrary invariant subspaces for arbitrary invariant subspaces of analytic functions in a half-plane. In other words, we study representation of all functions in an invariant subspace by the series of exponential monomials. These exponential monomials are eigenfunctions and adjoint functions for the differentiation operator in an invariant subspace. In the work we obtain a decomposition of an arbitrary invariant subspace of analytic functions into a sum of two invariant subspaces. We prove that the invariant subspace in an unbounded domain can be represented as a sum of two invariant subspaces. Their spectra correspond to a bounded and unbounded parts of a convex domain. On the base of this result we obtain a simple geometric criterion of the fundamental principle for an invariant subspace of analytic functions in a half-plane. It is formulated just in terms of the Krisvosheev condensation index for the sequence of exponents of the mentioned exponential monomials.

MSC:

30D10 Representations of entire functions of one complex variable by series and integrals
PDFBibTeX XMLCite
Full Text: DOI MNR

References:

[1] A. F. Leont’ev, Entire functions. Exponential series, Nauka, M., 1983 (in Russian)
[2] I. F. Krasičkov-Ternovskiĭ, “Invariant subspaces of analytic functions. I. Spectral analysis on convex regions”, Math. USSR-Sb., 16:4 (1972), 471-500 · Zbl 0253.46041 · doi:10.1070/SM1972v016n04ABEH001436
[3] I. F. Krasičkov-Ternovskiĭ, “Invariant subspaces of analytic functions. II. Spectral synthesis of convex domains”, Math. USSR-Sb., 17:1 (1972), 1-29 · Zbl 0253.46042 · doi:10.1070/SM1972v017n01ABEH001488
[4] A. A. Goldberg, B. Ya. Levin, I. V. Ostrovskii, “Entire and meromorphic functions”, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 85, 1991, 5-185 (in Russian) · Zbl 0734.30001
[5] A. S. Krivosheev, “A fundamental principle for invariant subspaces in convex domains”, Izv. Math., 68:2 (2004), 291-353 · Zbl 1071.30024 · doi:10.1070/IM2004v068n02ABEH000476
[6] O. A. Krivosheeva, A. S. Krivosheev, “A criterion for the fundamental principle to hold for invariant subspaces on bounded convex domains in the complex plane”, Funct. Anal. Appl., 46:4 (2012), 249-261 · Zbl 1274.46063 · doi:10.1007/s10688-012-0033-1
[7] A. S. Krivosheev, O. A. Krivosheeva, “A basis in an invariant subspace of analytic functions”, Sb. Math., 204:12 (2013), 1745-1796 · Zbl 1294.30057 · doi:10.1070/SM2013v204n12ABEH004359
[8] A. S. Krivosheev, O. A. Krivosheeva, “Fundamental principle and a basis in invariant subspaces”, Math. Notes, 99:5 (2016), 685-696 · Zbl 1353.30003 · doi:10.1134/S0001434616050072
[9] A. S. Krivosheev, O. A. Krivosheyeva, “A basis in invariant subspace of entire functions”, St. Petersburg Math. J., 27:2 (2016), 273-316 · Zbl 1335.30012 · doi:10.1090/spmj/1387
[10] A. S. Krivosheyev, O. A. Krivosheyeva, “A closedness of set of Dirichlet series sums”, Ufa Math. J., 5:3 (2013), 94-117 · doi:10.13108/2013-5-3-94
[11] O. A. Krivosheyeva, A. S. Krivosheyev, “A representation of functions from an invariant subspace with almost real spectrum”, St. Petersburg Math. J., 29:4 (2018), 603-641 · Zbl 1392.30002 · doi:10.1090/spmj/1509
[12] A. F. Leont’ev, “Sequences of exponential polynomials”, Nauka, M., 1980 (in Russian) · Zbl 0478.30005
[13] B. Ya. Levin, Distribution of zeros of entire functions, Amer. Math. Soc., Providence, RI, 1980 · Zbl 0152.06703
[14] B. N. Khabibullin, “On the growth of entire functions of exponential type along the imaginary axis”, Math. USSR-Sb., 67:1 (1990), 149-163 · Zbl 0699.30023 · doi:10.1070/SM1990v067n01ABEH003735
[15] P. Malliaven, L. Rubel, “On small entire functions of exponential type with given zeros”, Bull. Soc. Math. France, 89 (1961), 175-201 · Zbl 0126.29002 · doi:10.24033/bsmf.1564
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.