Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Quasitriangular + small compact
= strongly irreducible


Author: You Qing Ji
Journal: Trans. Amer. Math. Soc. 351 (1999), 4657-4673
MSC (1991): Primary 47A10, 47A55, 47A58
DOI: https://doi.org/10.1090/S0002-9947-99-02307-7
Published electronically: July 20, 1999
MathSciNet review: 1603910
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $T$ be a bounded linear operator acting on a separable infinite dimensional Hilbert space. Let $\epsilon $ be a positive number. In this article, we prove that the perturbation of $T$ by a compact operator $K$ with $\Vert K\Vert <\epsilon $ can be strongly irreducible if $T$ is a quasitriangular operator with the spectrum $\sigma (T)$ connected. The Main Theorem of this article nearly answers the question below posed by D. A. Herrero.

Suppose that $T$ is a bounded linear operator acting on a separable infinite dimensional Hilbert space with $\sigma (T)$ connected. Let $\epsilon >0$ be given. Is there a compact operator $K$ with $\Vert K\Vert <\epsilon $ such that $T+K $ is strongly irreducible?


References [Enhancements On Off] (What's this?)

  • [1] C. Apostol, C. Foias and D. Voiculescu, Some results on nonquasitriangular operators II, Rev. Roumaine Math. Pures et Appl. 18 (1973), 159-181. MR 48:12109a
  • [2] L. A. Fialkow, A note on the range of the operator $X\rightarrow AX-XB$, Illinois J. Math. 25 (1981), 112-124. MR 84b:47021
  • [3] D. A. Herrero, Approximation of Hilbert space operators, I, 2nd ed., Pitman Research Notes in Math., 224. Longman Group UK Limited, 1989. MR 91k:47002
  • [4] -, The diagonal entries in the formula `quasitriangular-compact = triangular', and restrictions of quasitriangularity, Trans. Amer. Math. Soc. 298 (1986), 1-42. MR 88c:47022
  • [5] -, Spectral pictures of operators in the Cowen-Douglas class ${\mathcal B}_n(\Omega )$ and its closure, J. Operator Theory 18 (1987), 213-222. MR 89b:47032
  • [6] D. A. Herrero, C. L. Jiang, Limits of strongly irreducible operators and the Riesz decomposition theorem, Michigan. Math. J. 37 (1990), 283-291. MR 91k:47035
  • [7] Y. Q. Ji, C. L. Jiang and Z. Y. Wang, The strongly irreducible operators in the nest algebras, Integral Equations and Operator Theory 28 (1997), 28-44. MR 98b:47057
  • [8] -, Essentially normal + small compact = strongly irreducible, Chinese Math. Ann. Series B 18 (1997), 485-494. MR 98i:47011
  • [9] - The $({\mathcal U}+{\mathcal K})$-orbit of essentially normal operators and compact perturbation of strongly irreducible operators, Functional Analysis in China, Math. and Its Applications, Vol. 356, Kluwer, Dordrecht, 1996, pp. 307-314. MR 97h:47013
  • [10] C. L. Jiang, S. H. Sun and Z. Y. Wang, Essentially normal operator + compact operator = strongly irreducible operator, Trans. Amer. Math. Soc. 349 (1997), 217-233. MR 97h:47012
  • [11] C. L. Jiang, S. Power and Z. Y. Wang, Biquasitriangular + small compact = strongly irreducible, J. London Math. Soc. (to appear)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 47A10, 47A55, 47A58

Retrieve articles in all journals with MSC (1991): 47A10, 47A55, 47A58


Additional Information

You Qing Ji
Affiliation: Department of Mathematics, Jilin University, Changchun 130023, P.R. China

DOI: https://doi.org/10.1090/S0002-9947-99-02307-7
Keywords: Weyl spectrum, index, strongly irreducible, quasitriangular
Received by editor(s): May 23, 1997
Published electronically: July 20, 1999
Additional Notes: This work is supported by MCSEC
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society