On the invertibility of one integral operator
DOI:
https://doi.org/10.52737/18291163-2022.14.6-1-10Keywords:
Integral operator, exponential integral function, $\mathcal{L}$-Wiener-Hopf operatorAbstract
The present paper considers an integral operator defined on the entire real axis, which differs from the Hilbert transform with terms where kernels are constructed using integral exponential functions. The considered operator has similar properties with respect to the Hilbert transform. The form of the inverse operator is obtained.
References
R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc., 176 (1973), pp. 227-251. https://doi.org/10.1090/s0002-9947-1973-0312139-8
A. Böttcher, Yu. I. Karlovich, and I. M. Spitkovsky, Convolution operators and factorization of almost periodic matrix functions, Operator Theory: Advances and Applications, Birkhäuser, Basel, 131 (2002). https://doi.org/10.1007/978-3-0348-8152-4
A. G. Kamalyan and I. M. Spitkovsky, On the Fredholm property of a class of convolution-type operators, Math. Notes, 104 (2018), pp. 404-416. https://doi.org/10.1134/s0001434618090080
A. G. Kamalyan, M. I. Karakhanyan, and A. H. Hovhannisyan, On a class of L-Wiener-Hopf operators, J. Contemp. Math. Anal., 53 (2018), pp. 134-138. https://doi.org/10.3103/s1068362318030032
D. Hasanyan, A. Kamalyan, M. Karakhanyan, and I. M. Spitkovsky, Integral operators of the L-convolution type in the case of a reflectionless potential, in Modern Methods in Operator Theory and Harmonic Analysis. OTHA 2018, ed. by A. Karapetyants, V. Kravchenko, and E. Liflyand, Springer Proceedings in Mathematics & Statistics, Springer, Cham, 291 (2019), pp. 175-197. https://doi.org/10.1007/978-3-030-26748-3_11
H. A. Asatryan, A. G. Kamalyan, and M. I. Karakhanyan, On L-convolution type operators with semialmost periodic symbols, Rep. Natl. Acad. Sci. Arm., 119 (2019), no. 1, pp. 22-28.
H. A. Asatryan, A. G. Kamalyan, and M. I. Karakhanyan, On a class of integro-difference equations, Rep. Natl. Acad. Sci. Arm., 119 (2019), no. 2, pp. 103-109.
A. G. Kamalyan and G. A. Kirakosyan, L-Wiener-Hopf operators in weighted spaces in case of reflectionless potential, Journal of Contemporary Mathematical Analysis, 57 (2022), no. 2, pp. 112-121.
H. S. Grigoryan, A. G. Kamalyan, and G. A. Kirakosyan, L-Wiener-Hopf operators with piecewise continuous matrix-valued symbol on Lebesgue spaces with power weight, Reports of NAS RA, 121 (2021), no. 4, pp. 259-264 (in Russian).
L. D. Faddeev, Inverse problem of the quantum scattering theory, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat., VINITI, Moscow, 3 (1974), pp. 93-180.
V. A. Marchenko, Sturm-Liouville operators and their applications, Naukova Dumka, Kiev, (1977).
V. Yurko, Introduction to the theory of inverse spectral problems, Fizmatlit, Moscow, (2007).
P. L. Bhatnagar, Nonlinear waves in one-dimensional dispersive systems, Oxford Univ. Press, (1979).
V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of solitons: method of inverse problem, Nauka, Moscow, (1980).
F. Calogero and A. Degasperis, Spectral transform and solitons, North-Holland, Amsterdam, (1982).
O. D. Khvolson, Grundzüge einer mathematischen theorie der inneren diffusion des lichtes, Proceedings of the Petersburg Academy of Science, 33 (1890), p. 221.
E. A. Milne, Radiative equilibrium in the outer layers of a star, Monthly Notices of the RAS, 81 (1921), pp. 361-375. https://doi.org/10.1093/mnras/81.5.361
V. V. Sobolev, Radiative transfer in the atmospheres of stars and planets, Moscow, Gostexizdat, (1956).
M. G. Krein, Integral equations on a half-linement with a kernel depending upon the difference of the arguments, Amer. Math. Soc. Transl., 22 (1962), pp. 163-238.
F. D. Gakhov, Boundary value problems, Dover Publ. Inc., New York, (1990).
Downloads
Published
Issue
Section
License
Copyright (c) 2022 Armenian Journal of Mathematics
This work is licensed under a Creative Commons Attribution 4.0 International License.
