Abstract
We consider the sequence of the finite sections R n AR n of a band-dominated operator A on l 2(ℤ) with almost periodic coefficients. Our main result says that if the compressions of A onto ℤ+ and ℤ− are invertible, then there is a distinguished subsequence of (R n AR n) which is stable. Moreover, this subsequence proves to be fractal, which allows us to establish the convergence in the Hausdorff metric of the singular values and pseudoeigenvalues of the finite section matrices.
The first two authors are supported by CONACYT project 43432.
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References
W. Arveson, Improper filtrations for C*-algebras: Spectra of unilateral tridiagonal operators. — Acta Sci. Math. (Szeged) 57(1993), 11–24.
W. Arveson, C*-algebras and numerical linear algebra. — J. Funct. Anal. 122(1994), 333–360.
W. Arveson, The role of C*-algebras in infinite dimensional numerical linear algebra. — Contemp. Math. 167(1994), 115–129.
F.P. Boca, Rotation C*-Algebras and Almost Mathieu Operators. — Theta Series in Advanced Mathematics 1, The Theta Foundation, Bucharest 2001.
A. Böttcher, Infinite matrices and projection methods.-In: P. Lancaster (Ed.), Lectures on Operator Theory and its Applications, Fields Institute Monographs Vol. 3, Amer. Math. Soc., Providence, Rhode Island 1995, 1–72.
A. Böttcher, S.M. Grudsky, B. Silbermann, Norms of inverses, spectra, and pseudospectra of large truncated Wiener-Hopf operators and Toeplitz matrices. — New York J. Math. 3(1997), 1–31.
A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators. — Akademie-Verlag, Berlin 1989 and Springer-Verlag, Berlin, Heidelberg, New York 1990.
A. Böttcher, B. Silbermann, Introduction to Large Truncated Toeplitz Matrices. — Springer-Verlag, Berlin, Heidelberg 1999.
N. Brown, AF embeddings and the numerical computation of spectra in irrational rotation algebras. — Preprint 2004.
I. Gohberg, I. Feldman, Convolution Equations and Projection Methods for Their Solution. — Nauka, Moskva 1971 (Russian, Engl. transl.: Amer. Math. Soc. Transl. of Math. Monographs, Vol. 41, Providence, Rhode Island, 1974).
R. Hagen, S. Roch, B. Silbermann, C*-Algebras and Numerical Analysis. — Marcel Dekker, Inc., New York, Basel 2001.
A.V. Kozak, I.B. Simonenko, Projection methods for the solution of multidimensional convolution equations. — Sib. Mat. Zh. 21(1980), 2, 119–127 (Russian).
S. Prössdorf, B. Silbermann, Numerical Analysis for Integral and Related Operator Equations. — Akademie-Verlag, Berlin, 1991, and Birkhäuser Verlag, Basel, Boston, Stuttgart 1991.
V.S. Rabinovich, S. Roch, B. Silbermann, Fredholm theory and finite section method for band-dominated operators. — Integral Equations Oper. Theory 30(1998), 4, 452–495.
V.S. Rabinovich, S. Roch, B. Silbermann, Algebras of approximation sequences: Finite sections of band-dominated operators. — Acta Appl. Math. 65(2001), 315–332.
V.S. Rabinovich, S. Roch, B. Silbermann, Limit Operators and Their Applications in Operator Theory. — Operator Theory: Adv. and Appl. 150, Birkhäuser Verlag, Basel, Boston, Berlin 2004.
S. Roch, Spectral approximation of Wiener-Hopf operators with almost periodic generating function. — Numer. Funct. Anal. Optimization 21(2000), 1–2, 241–253.
S. Roch, Finite sections of band-dominated operators. — Preprint 2355 TU Darmstadt, July 2004, 98 p., submitted to Memoirs Amer. Math. Soc.
A. Rogozhin, B. Silbermann, On the approximation numbers for the finite sections of block Toeplitz matrices. — Submitted to London Math. Soc.
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Dedicated to I.B. Simonenko on occasion of his seventieth birthday.
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Rabinovich, V.S., Roch, S., Silbermann, B. (2006). Finite Sections of Band-dominated Operators with Almost Periodic Coefficients. In: Erusalimsky, Y.M., Gohberg, I., Grudsky, S.M., Rabinovich, V., Vasilevski, N. (eds) Modern Operator Theory and Applications. Operator Theory: Advances and Applications, vol 170. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7737-3_12
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DOI: https://doi.org/10.1007/978-3-7643-7737-3_12
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