Quasidiagonality of direct sums of weighted shifts
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- by Sivaram K. Narayan PDF
- Trans. Amer. Math. Soc. 332 (1992), 757-774 Request permission
Abstract:
Let $\mathcal {H}$ be a separable Hilbert space. A bounded linear operator $A$ defined on $\mathcal {H}$ is said to be quasidiagonal if there exists a sequence $\{ {P_n}\}$ of projections of finite rank such that ${P_n} \to I$ strongly and $\left \| A{P_n} - {P_n}A\right \| \to 0$ as $n \to \infty$. We give a necessary and sufficient condition for a finite direct sum of weighted shifts to be quasidiagonal. The condition is stated using a marked graph (a graph with a $(0)$, $( + )$ or $( - )$ attached to its vertices) that can be associated with the direct sum.References
-
C. Apostol, C. Foias, and D. Voiculescu, Some results on non-quasitriangular operators, IV, Rev. Roumaine Math. Pures Appl. 18 (1973), 487-514.
- Lowell W. Beineke and Frank Harary, Consistency in marked digraphs, J. Math. Psych. 18 (1978), no.Β 3, 260β269. MR 522390, DOI 10.1016/0022-2496(78)90054-8
- Lowell W. Beineke and Frank Harary, Consistent graphs with signed points, Riv. Mat. Sci. Econom. Social. 1 (1978), no.Β 2, 81β88 (English, with Italian summary). MR 573718, DOI 10.1007/BF02631374
- J. A. Bondy and U. S. R. Murty, Graph theory with applications, American Elsevier Publishing Co., Inc., New York, 1976. MR 0411988
- L. G. Brown, The universal coefficient theorem for $\textrm {Ext}$ and quasidiagonality, Operator algebras and group representations, Vol. I (Neptun, 1980) Monogr. Stud. Math., vol. 17, Pitman, Boston, MA, 1984, pp.Β 60β64. MR 731763 β, Generalized crossed products of ${C^{\ast }}$-algebras (in preparation).
- John B. Conway, A course in functional analysis, Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1985. MR 768926, DOI 10.1007/978-1-4757-3828-5
- Don Hadwin, Strongly quasidiagonal $C^*$-algebras, J. Operator Theory 18 (1987), no.Β 1, 3β18. With an appendix by Jonathan Rosenberg. MR 912809
- P. R. Halmos, Quasitriangular operators, Acta Sci. Math. (Szeged) 29 (1968), 283β293. MR 234310
- P. R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887β933. MR 270173, DOI 10.1090/S0002-9904-1970-12502-2
- Paul Richard Halmos, A Hilbert space problem book, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 17, Springer-Verlag, New York-Berlin, 1982. MR 675952
- Domingo A. Herrero, Approximation of Hilbert space operators. Vol. I, Research Notes in Mathematics, vol. 72, Pitman (Advanced Publishing Program), Boston, MA, 1982. MR 676127
- Glenn R. Luecke, A note on quasidiagonal and quasitriangular operators, Pacific J. Math. 56 (1975), no.Β 1, 179β185. MR 374963
- Carl M. Pearcy, Some recent developments in operator theory, Regional Conference Series in Mathematics, No. 36, American Mathematical Society, Providence, R.I., 1978. MR 0487495
- Allen L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp.Β 49β128. MR 0361899 R. A. Smucker, Quasidiagonal and quasitriangular operators, Dissertation, Indiana Univ., 1973.
- Russell Smucker, Quasidiagonal weighted shifts, Pacific J. Math. 98 (1982), no.Β 1, 173β182. MR 644948
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 332 (1992), 757-774
- MSC: Primary 47B37; Secondary 47A66
- DOI: https://doi.org/10.1090/S0002-9947-1992-1012511-9
- MathSciNet review: 1012511