Quasidiagonality of direct sums of weighted shifts

Author:
Sivaram K. Narayan

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

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a separable Hilbert space. A bounded linear operator defined on is said to be *quasidiagonal* if there exists a sequence of projections of finite rank such that strongly and as .

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 , or attached to its vertices) that can be associated with the direct sum.

**[1]**C. Apostol, C. Foias, and D. Voiculescu,*Some results on non-quasitriangular operators*, IV, Rev. Roumaine Math. Pures Appl.**18**(1973), 487-514.**[2]**Lowell W. Beineke and Frank Harary,*Consistency in marked digraphs*, J. Math. Psych.**18**(1978), no. 3, 260–269. MR**522390**, https://doi.org/10.1016/0022-2496(78)90054-8**[3]**Lowell W. Beineke and Frank Harary,*Consistent graphs with signed points*, Riv. Mat. Sci. Econom. Social.**1**(1978), no. 2, 81–88 (English, with Italian summary). MR**573718**, https://doi.org/10.1007/BF02631374**[4]**J. A. Bondy and U. S. R. Murty,*Graph theory with applications*, American Elsevier Publishing Co., Inc., New York, 1976. MR**0411988****[5]**L. G. Brown,*The universal coefficient theorem for 𝐸𝑥𝑡 and quasidiagonality*, Operator algebras and group representations, Vol. I (Neptun, 1980) Monogr. Stud. Math., vol. 17, Pitman, Boston, MA, 1984, pp. 60–64. MR**731763****[6]**-,*Generalized crossed products of*-*algebras*(in preparation).**[7]**John B. Conway,*A course in functional analysis*, Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1985. MR**768926****[8]**Don Hadwin,*Strongly quasidiagonal 𝐶*-algebras*, J. Operator Theory**18**(1987), no. 1, 3–18. With an appendix by Jonathan Rosenberg. MR**912809****[9]**P. R. Halmos,*Quasitriangular operators*, Acta Sci. Math. (Szeged)**29**(1968), 283–293. MR**0234310****[10]**P. R. Halmos,*Ten problems in Hilbert space*, Bull. Amer. Math. Soc.**76**(1970), 887–933. MR**0270173**, https://doi.org/10.1090/S0002-9904-1970-12502-2**[11]**Paul Richard Halmos,*A Hilbert space problem book*, 2nd ed., Graduate Texts in Mathematics, vol. 19, Springer-Verlag, New York-Berlin, 1982. Encyclopedia of Mathematics and its Applications, 17. MR**675952****[12]**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****[13]**Glenn R. Luecke,*A note on quasidiagonal and quasitriangular operators*, Pacific J. Math.**56**(1975), no. 1, 179–185. MR**0374963****[14]**Carl M. Pearcy,*Some recent developments in operator theory*, American Mathematical Society, Providence, R.I., 1978. Regional Conference Series in Mathematics, No. 36. MR**0487495****[15]**Allen L. Shields,*Weighted shift operators and analytic function theory*, Topics in operator theory, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. Math. Surveys, No. 13. MR**0361899****[16]**R. A. Smucker,*Quasidiagonal and quasitriangular operators*, Dissertation, Indiana Univ., 1973.**[17]**Russell Smucker,*Quasidiagonal weighted shifts*, Pacific J. Math.**98**(1982), no. 1, 173–182. MR**644948**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1992-1012511-9

Keywords:
Quasidiagonality,
weighted shifts,
marked graphs,
crossed products of -algebras

Article copyright:
© Copyright 1992
American Mathematical Society