The semigroup generated by a similarity orbit or a unitary orbit of an operator
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- by C. K. Fong and A. R. Sourour PDF
- Proc. Amer. Math. Soc. 131 (2003), 3203-3210 Request permission
Abstract:
Let $T$ be an invertible operator that is not a scalar modulo the ideal of compact operators. We show that the multiplicative semigroup generated by the similarity orbit of $T$ is the group of all invertible operators. If, in addition, $T$ is a unitary operator, then the multiplicative semigroup generated by the unitary orbit of $T$ is the group of all unitary operators.References
- Arlen Brown and Carl Pearcy, Structure of commutators of operators, Ann. of Math. (2) 82 (1965), 112–127. MR 178354, DOI 10.2307/1970564
- L. G. Brown, R. G. Douglas, and P. A. Fillmore, Unitary equivalence modulo the compact operators and extensions of $C^{\ast }$-algebras, Proceedings of a Conference on Operator Theory (Dalhousie Univ., Halifax, N.S., 1973) Lecture Notes in Math., Vol. 345, Springer, Berlin, 1973, pp. 58–128. MR 0380478
- Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
- R. G. Douglas, Banach algebra techniques in operator theory, Academic Press, New York, 1972. 50:14335
- P. A. Fillmore, J. G. Stampfli, and J. P. Williams, On the essential numerical range, the essential spectrum, and a problem of Halmos, Acta Sci. Math. (Szeged) 33 (1972), 179–192. MR 322534
- C. K. Fong, C. R. Miers, and A. R. Sourour, Lie and Jordan ideals of operators on Hilbert space, Proc. Amer. Math. Soc. 84 (1982), no. 4, 516–520. MR 643740, DOI 10.1090/S0002-9939-1982-0643740-0
- C. K. Fong and A. R. Sourour, The group generated by unipotent operators, Proc. Amer. Math. Soc. 97 (1986), no. 3, 453–458. MR 840628, DOI 10.1090/S0002-9939-1986-0840628-0
- L. Grunenfelder, M. Omladic, H. Radjavi, and A. Sourour, Semigroups generated by similarity orbits, Semigroup Forum 62 (2001), no. 3, 460–472. MR 1831467, DOI 10.1007/s002330010029
- Paul R. Halmos and Shizuo Kakutani, Products of symmetries, Bull. Amer. Math. Soc. 64 (1958), 77–78. MR 100225, DOI 10.1090/S0002-9904-1958-10156-1
- N. Christopher Phillips, Every invertible Hilbert space operator is a product of seven positive operators, Canad. Math. Bull. 38 (1995), no. 2, 230–236. MR 1335103, DOI 10.4153/CMB-1995-033-9
- H. Radjavi, The group generated by involutions, Proc. Roy. Irish Acad. Sect. A 81 (1981), no. 1, 9–12. MR 635572
- Heydar Radjavi and Peter Rosenthal, Invariant subspaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77, Springer-Verlag, New York-Heidelberg, 1973. MR 0367682
- Pei Yuan Wu, The operator factorization problems, Linear Algebra Appl. 117 (1989), 35–63. MR 993030, DOI 10.1016/0024-3795(89)90546-6
Additional Information
- C. K. Fong
- Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6
- A. R. Sourour
- Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada V8W 3P4
- Email: sourour@math.uvic.ca
- Received by editor(s): November 22, 2000
- Received by editor(s) in revised form: May 17, 2002
- Published electronically: May 9, 2003
- Additional Notes: This research was supported in part by an NSERC grant.
- Communicated by: David R. Larson
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 3203-3210
- MSC (2000): Primary 47D03; Secondary 20F38
- DOI: https://doi.org/10.1090/S0002-9939-03-06910-7
- MathSciNet review: 1992861