Spectral resolutions for non-self-adjoint block convolution operators

• Autorzy: Zalot Ewelina

Identyfikatory

DOI

10.7494/OpMath.2022.42.3.459

• Języki publikacji: EN

Abstrakty

EN

The paper concerns the spectral theory for a class of non-self-adjoint block convolution operators. We mainly discuss the spectral representations of such operators. It is considered the general case of operators defined on Banach spaces. The main results are applied to periodic Jacobi matrices.

Słowa kluczowe

EN

spectral operators; chains; triangular decomposition; Laurent operators; Jacobi matrices

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2022
• Tom: Vol. 42, no. 3
• Strony: 459--487
• Opis fizyczny: Bibliogr. 21 poz.

Twórcy

autor

Zalot Ewelina

• AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Kraków, Poland

• https://orcid.org/0000-0002-1050-6756

Bibliografia

• [1] N.I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Dover Publ. Inc., New York, 2020.
• [2] M.S. Brodskii, Triangular and Jordan Representations of Linear Operators, Transl. Math., Monographs, vol. 32, Amer. Math. Soc., Providence, R.I., 1971.
• [3] V.M. Brodskii, I.Ts. Gokhberg, M.G. Krein, General theorems on triangular representations of linear operators and multiplications of their characteristic functions, Funktsional. Anal. i Prilozhen. 3:4 (1969), 1–27; Transl. in: Funct. Anal. Appl. 3:4 (1969), 255–276.
• [4] P.A. Cojuhari, On the spectrum of a class of block Jacobi matrices, operator theory, structured matrices, and dilations, Theta Ser. Adv. Math., Bucharest (2007), 137–152.
• [5] P.A. Cojuhari, Discrete spectrum in the gaps for perturbations of periodic Jacobi matrices, J. Comput. Appl. Math. 225 (2009), no. 2, 374–386.
• [6] N. Dunford, Spectral theory, II. Resolutions of the identity, Pacific J. Math. 2 (1952), 559–614.
• [7] N. Dunford, Spectral operators, Pacific J. Math. 4 (1954), 321–354.
• [8] N. Dunford, A survey of the theory of spectral operators, Bull. Amer. Math. Soc. 64 (1958), 217–274.
• [9] Ya.L. Geronimus, Orthogonal Polynomials, Amer. Math. Soc. Transl., Ser. 2, vol. 108, (1977).
• [10] I.M. Glazman, Direct methods of qualitative spectral analysis of singular differential operators, Fiz.-Mat. (1963) [in Russian], English by Israel Program for Scientific Translation, 1965.
• [11] I. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of Linear Operators, vol. II, Birkhäuser, Basel, 1993.
• [12] I. Gohberg, M. Krein, Theory of Vollterra Operators in Hilbert Space and its Applications, “Nauka”, Moscow, 1967; English transl., Transl. Math., Monographs, vol. 24, Amer. Math. Soc., Providence, R.I., 1970.
• [13] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, 1985.
• [14] M.S. Livshits, On the spectral decomposition of linear non-selfadjoint operators, Mat. Sb. 34 (76) (1954), 145–199 [in Russian]; English transl. in: Amer. Math. Soc. Transl. (2) 5 (1957), 67–114.
• [15] W. Magnus, S. Winkler, Hill’s Equation, Interscience Publ., New York, London, Sydney, 1966.
• [16] P.B. Naïman, On the theory of periodic and limit-periodic Jacobi matrices, Dokl. Akad. Nauk 143 (1962), no. 2, 277–279 [in Russian].
• [17] P.B. Naïman, On the spectral theory of non-hermitian periodic Jacobi matrices, Dopov. Akad. Nauk Ukr. RSR 10 (1963), 1307–1311 [in Ukrainian].
• [18] P.B. Naïman, On the spectral theory of the non-symetric periodic Jacobi matrices, Notes of the Faculty of Math. and Mech. of Kharkov’s State University and of Kharkov’s Math. Society 30 (1964), 138–151 [in Russian].
• [19] A.M. Ostrowski, Solution of Equations and Systems of Equations, vol. 9 of Pure and Applied Mathematics, Academic Press, New York, London 1960.
• [20] F. Rellich, Perturbation Theory of Eigenvalue Problems, CRC Press, New York, London, Paris, 1969.
• [21] E. Zalot, W. Majdak, Spectral representations for a class of banded Jacobi-type matrices, Opuscula Math. 34 (2014), no. 4, 871–887.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).