Mathematics of Computation

Author(s): Nathanial P. Brown.
Journal: Math. Comp. 76 (2007), 339-360.
MSC (2000): Primary 65J10, 46N40
Posted: August 31, 2006
Retrieve article in: PDF

Abstract: Quasi-diagonal operators on a Hilbert space are a large and important class (containing all self-adjoint operators for instance). They are also perfectly suited for study via the finite section method (a particular Galerkin method). Indeed, the very definition of quasi-diagonality yields finite sections with good convergence properties. Moreover, simple operator theory techniques yield estimates on certain rates of convergence. In the case of quasi-diagonal band operators both the finite sections and rates of convergence are explicitly given.

1.: W. Arveson, The role of C-algebras in infinite dimensional numerical linear algebra, C-algebras: 1943-1993 (San Antonio, TX, 1993), 114-129, Contemp. Math. 167, Amer. Math. Soc., Providence, RI, 1994. MR 1292012 (95i:46084)
2.: I.D. Berg, An extension of the Weyl-von Neumann theorem to normal operators, Trans. Amer. Math. Soc. 160 (1971), 365-371. MR 0283610 (44:840)
3.: A. Böttcher, C-algebras in numerical analysis, Irish Math. Soc. Bull. No. 45 (2000), 57-133. MR 1832325 (2002b:46116)
4.: N.P. Brown, On quasi-diagonal C-algebras, Adv. Stud. Pure Math., vol. 38, Math. Soc. Japan, Tokyo, 2004, pp. 19-64. MR 2059800 (2005a:46107)
5.: N.P. Brown, Invariant means and finite representation theory of C-algebras, Mem. Amer. Math. Soc. (accepted).
6.: N.P. Brown, AF embeddings and the numerical computation of spectra in irrational rotation algebras, preprint.
7.: N.P. Brown, Herrero's approximation problem for quasi-diagonal operators, J. Funct. Anal. 186 (2001), 360-365. MR 1864827 (2002j:47034)
8.: M. Dadarlat, On the approximation of quasi-diagonal C-algebras, J. Funct. Anal. 167 (1999), 69-78.MR 1710645 (2000f:46069)
9.: K.R. Davidson, C-algebras by example, Fields Institute Monographs 6, American Mathematical Society, Providence, RI, 1996. MR 1402012 (97i:46095)
10.: M. Embree and L.N. Trefethen, Pseudospectra Gateway: http://www.comlab.ox.ac.uk/ pseudospectra.
11.: R. Hagen, S. Roch and B. Silbermann, C-algebras and numerical analysis. Monographs and Textbooks in Pure and Applied Mathematics, 236. Marcel Dekker, Inc., New York, 2001.MR 1792428 (2002g:46133)
12.: G.H. Hardy and E.M. Wright, An introduction to the theory of numbers. Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979. MR 0568909 (81i:10002)
13.: P.R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887-933. MR 0270173 (42:5066)
14.: E. Kirchberg, Exact C-algebras, tensor products and the classification of purely infinite algebras, Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Zurich, 1994), 943-954. MR 1403994 (97g:46074)
15.: S. Narayan, Quasi-diagonality of direct sums of weighted shifts, Trans. Amer. Math. Soc. 332 (1992), 757-774.MR 1012511 (92j:47048)
16.: R. Smucker, Quasi-diagonal weighted shifts, Pacific J. Math. 98 (1982), 173-182. MR 0644948 (83c:47045)
17.: S. Szarek, An exotic quasi-diagonal operator, J. Funct. Anal. 89 (1990), 274-290. MR 1042211 (91e:47015)
18.: D.V. Voiculescu, Around quasi-diagonal operators, Integr. Equ. and Op. Thy. 17 (1993), 137-149. MR 1220578 (94e:47029)
19.: D.V. Voiculescu, A note on quasi-diagonal operators, Operator Theory: Advances and Applications, Vol. 32, Birkhauser Verlag, Basel, 1988, 265-274. MR 0951964 (89h:47029)
20.: S. Wassermann, Exact C-algebras and related topics, Lecture Notes Series no. 19, GARC, Seoul National University, 1994.MR 1271145 (95b:46081)

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