Abstract
The topics of this paper are Fredholm properties and the applicability of the finite section method for band operators onl p-spaces as well as for their norm limits which we call band-dominated operators. The derived criteria will be established in terms of the limit operators of the given band-dominated operator. After presenting the general theory, we present its specifications to concrete classes of band-dominated operators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
W. Arveson,C *-algebras and numerical linear algebra. — Journal Fctl. Analysis122(1994), 333–360.
W. Arveson, The role ofC *-algebras in infinite dimensional numerical linear algebra. — Contemporary Mathematics167(1994), 115–129.
A. Ben-Artzi, I. Gohberg, Fredholm properties of band matrices and dichotomy. —In: Topics in operator theory, The Constantin Apostol memorial issue (ed.I. Gohberg, Operator Theory: Advances and Appl.32, Birkhäuser Verlag, Basel 1988, 37–52.
A. Böttcher, B. Silbermann, Analysis of Toeplitz operators. — Akademie-Verlag, Berlin 1989 and Springer-Verlag, Berlin, Heidelberg, New York 1990.
L. M. Brekhovskikh, O. A. Godin, Acoustics in stratified media. — Nauka, Moskva 1989 (Russian).
R. G. Douglas, On the invertibility of a class of Toeplitz operators on the quarter plane. — Ind. Univ. Math. J.21(1972), 1031–1036.
R. G. Douglas, R. Howe, On the invertibility of a class of Toeplitz operators on the quarter plane. — Transactions AMS158(1971), 203–217.
J. Farvard, Sur les equations differentiales a coefficients presque periodicues. — Acta Math.51(1927), 31–81.
I. M. Gelfand, D. A. Raikov, G. E. Shilov, Commutative normed rings. — Goz. izd. Fiz.-Mat. Lit., Moskva 1960 (Russian).
I. Gohberg, I. Feldman, Convolution equations and projection methods for their solution. — Nauka, Moskva 1971 (Russian, Engl. transl.: Amer. Math. Soc. Transl. of Math. Monographs41, Providence, R.I., 1974).
I. Gohberg, M. A. Kaashoek, Projection method for Block Toeplitz operators with operator-valued symbols. — In: Toeplitz operators and related topics. Operator Theory: Advances and Applications 71, Birkhäuser Verlag, Basel-Boston-Berlin 1994, 79–104.
I. C. Gohberg, M. G. Krein, Systems of integral equations on the semi-axis with kernels depending on the difference of the arguments. — Uspekhi Mat. Nauk13(1958), 2, 3–72 (Russian).
R. Hagen, S. Roch, B. Silbermann, Spectral Theory of Approximation Methods for Convolution Equations. — Birkhäuser Verlag, Basel, Boston, Berlin 1995.
A. V. Kozak, A local principle in the theory of projection methods. — Dokl. AN SSSR212(1973), 6, 1287–1289 (Russian).
A. V. Kozak, I. B. Simonenko, Projection methods for solving discrete convolution equations. — Sib. Mat. J.20(1979), 6, 1249–1260 (Russian).
V. G. Kurbatov, Linear differential-difference operators. — Voronezh. Univ., Voronezh 1990 (Russian).
B. V. Lange, V. S. Rabinovich, On the Noether property of multidimensional discrete convolution operators. — Mat. zametki37(1985), 3, 407–421 (Russian).
B. V. Lange, V. S. Rabinovich, Pseudodifferential operators on ℝn and limit operators. — Mat. sbornik129(1986), 2, 175–185 (Russian).
V. A. Malyshev, Solution of discrete Wiener-Hopf equations on the quarter plane. —Dokl. AN SSSR187(1979), 1243–1246 (Russian).
E. M. Mukhamadijev, On the invertibility of differential operators in partial derivatives of elliptic type. — Dokl. AN SSSR205(1972), 6, 1292–1295 (Russian).
E. M. Mukhamadijev, On the normal solvability and the Noether property of elliptic operators on functions on ℝn. — Zap. nauchn. sem. LOMI110(1981), 120–140 (Russian).
S. Prössdorf, Einige Klassen singulärer Gleichungen. — Akademie-Verlag, Berlin 1974.Engl. transl.: North Holland 1978,Russian transl.: Mir, Moskva 1979.
V. S. Rabinovich, On the Fredholmness of pseudodifferential operators in the scale ofL 2,p spaces. — Sib. Mat. J.20(1988), 4, 149–161 (Russian).
S. Roch, B. Silbermann, Algebras of convolution operators and their image in the Calkin algebra. — Report R-MATH-05/90 des Karl-WeierstraßInstituts für Mathematik, Berlin 1990, 157 S.
I. B. Simonenko, A new general method of investigating linear operator equations of the singular integral equation type, I. — Izv. AN SSSR, Ser. Mat.,3(1965), 567–586 (Russian).
I. B. Simonenko, On multidimensional discrete convolutions. — Mat. issled.3(1968), 1, 108–127 (Russian).
V. Ju. Zavadskii, Grid methods for waveguides. — Nauka, Moskva 1986 (Russian).
Author information
Authors and Affiliations
Additional information
To the memory of Professor Mark Krein
Supported by the DFG grant 436 RUS/17/148/95
Supported by a DFG Heisenberg grant
Rights and permissions
About this article
Cite this article
Rabinovich, V.S., Roch, S. & Silbermann, B. Fredholm theory and finite section method for band-dominated operators. Integr equ oper theory 30, 452–495 (1998). https://doi.org/10.1007/BF01257877
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01257877