Spectral representations for a class of banded Jacobi-type matrices

• Autorzy: Zalot E. , Majdak W.

Identyfikatory

DOI

http://dx.doi.org/10.7494/OpMath.2014.34.4.871

• Języki publikacji: EN

Abstrakty

EN

We describe some spectral representations for a class of non-self-adjoint banded Jacobi-type matrices. Our results extend those obtained by P.B. Naïman for (two-sided infinite) periodic tridiagonal Jacobi matrices.

Słowa kluczowe

EN

spectral representation; Laurent operators; Jacobi matrices

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2014
• Tom: Vol. 34, no. 4
• Strony: 871--887
• Opis fizyczny: Bibliogr. 35 poz.

Twórcy

autor

Zalot E.

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

autor

Majdak W.

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

Bibliografia

• [1] A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, Springer-Verlag, Heidelberg, 1990.
• [2] C. Burnap, P.F. Zweifel, A note on the spectral theorem, Integral Equations Operator Theory 9 (1986), 305–324.
• [3] A. Cojuhari, J. Stochel, An integral representation for general operators in Banach spaces, Int. J. Pure Appl. Math. 54 (2009), 451–465.
• [4] A. Cojuhari, J. Stochel, A spectral representation for bounded non-selfadjoint operators, Complex Anal. Oper. Theory 6 (2012), 819–828.
• [5] 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.
• [6] P.A. Cojuhari, Discrete spectrum in the gaps for perturbations of periodic Jacobi matrices, J. Comput. Appl. Math. 225 (2009), 374–386.
• [7] I. Colojoara, Elemente de teorie specrala, Editura Academiei Rep. Soc. România, Bucaresti, 1968 [in Romanian].
• [8] I. Colojoara, C. Foias, Theory of Generalized Spectral Operators, Gordon and Breach, New York, 1968.
• [9] N. Dunford, Spectral theory, II. Resolutions of the identity, Pacific J. Math. 2 (1952), 559–614.
• [10] N. Dunford, Spectral operators, Pacific J. Math. 4 (1954), 321–354.
• [11] N. Dunford, A survey of the theory of spectral operators, Bull. Amer. Math. Soc. 64 (1958), 217–274.
• [12] N. Dunford, J.T. Schwartz, Linear Operators, Part III: Spectral Operators, Wiley Classics
• Library, Wiley, New York, 1988.
• [13] H. Dym, V. Katsnelson, Contributions of Issai Schur to Analysis, Studies in Memory of Issai Schur, Progr. in Math., 210, Birkhäuser, Boston, MA, 2003.
• [14] U. Fixman, Problems in spectral operators, Pacific J. Math. 9 (1959), 1029–1051.
• [15] H. Flaschka, Invariant subspaces of abstract multiplication operators, Indiana Univ. Math. J. 21 (1971), 413–418.
• [16] C. Foias, Spectral maximal spaces and decomposable operators in Banach space, Arch. Math. 14 (1963), 341–349.
• [17] T.A. Gillespie, A spectral theorem for Lp translations, J. London Math. Soc. 11 (1975), 499–508.
• [18] I.M. Glazman, Direct methods the qualitative spectral analysis of singular differential operators, Fiz.-Mat. (1963) [in Russian], English by Israel Program for Scientific Translation, 1965.
• [19] I.C. Gohberg, M. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Amer. Math. Soc., Providence, 1969.
• [20] Ju.I. Ljubic, On the spectrum of a representation of an Abelian topological group, Dokl. Akad. Nauk SSSR 200 (1971), 777–780.
• [21] Ju.I. Ljubic, V.I. Macaev, On the spectral theory of linear operators in Banach space, Dokl. Akad. Nauk SSSR 131 (1960), 21–23 [in Russian]; English transl. in: Soviet Math. Dokl. 1 (1960), 184–186.
• [22] Ju.I. Ljubic, V.I. Macaev, G.M. Fel’dman, On the separability of the spectrum of a representation of a locally compact Abelian group, Dokl. Akad. Nauk SSSR 201 (1971), 1282–1284 [in Russian].
• [23] F.D. Murnaghan, The Theory of Group Representations, Johns Hopkins, Baltimore, 1938.
• [24] P.B. Naïman, On the theory of periodic and limit-periodic Jacobi matrices, Dokl. Akad. Nauk 143 (1962), 277–279 [in Russian].
• [25] P.B. Naïman, On the spectral theory of non-hermitian periodic Jacobi matrices, Dopov. Akad. Nauk Ukr. RSR 10 (1963), 1307–1311 [in Ukrainian].
• [26] 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].
• [27] J. von Neumann, Zur Algebra der Funktionaloperatoren und Theorie der normalen Operatoren, Math. Ann. 102 (1929), 370–427 [in German].
• [28] J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, 1932 [in German], English transl.: Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, N.J., 1955.
• [29] H.H. Schaefer, A new class of spectral operators, Pacific J. Math. 9 (1959), 154–155.
• [30] H.H. Schaefer, Spectral measures in locally convex algebras, Acta Math. 107 (1962), 125–173.
• [31] H.H. Schaefer, Convex cones and spectral theory, Proc. Sympos. Pure Math., vol. VII, 451–471, Amer. Math. Soc., Providence, 1963.
• [32] B. Sz.-Nagy, C. Foias, Analyse Harmoniques des Opérateurs de L’espace de Hilbert, Akad. Kiad´ó, Budapest, 1967.
• [33] A. Turowicz, Matrix Theory, AGH, Kraków, 2005 [in Polish].
• [34] J. Wermer, The existence of an invariant subspace, Duke Math. J. 19 (1952), 615–622.
• [35] R. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 1980.