The semigroup generated by a similarity orbit or a unitary orbit of an operator

Authors:
C. K. Fong and A. R. Sourour

Journal:
Proc. Amer. Math. Soc. **131** (2003), 3203-3210

MSC (2000):
Primary 47D03; Secondary 20F38

Published electronically:
May 9, 2003

MathSciNet review:
1992861

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Abstract | References | Similar Articles | Additional Information

Abstract: Let 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 is the group of all invertible operators. If, in addition, is a unitary operator, then the multiplicative semigroup generated by the unitary orbit of is the group of all unitary operators.

**1.**Arlen Brown and Carl Pearcy,*Structure of commutators of operators*, Ann. of Math. (2)**82**(1965), 112–127. MR**0178354****2.**L. G. Brown, R. G. Douglas, and P. A. Fillmore,*Unitary equivalence modulo the compact operators and extensions of 𝐶*-algebras*, Proceedings of a Conference on Operator Theory (Dalhousie Univ., Halifax, N.S., 1973) Springer, Berlin, 1973, pp. 58–128. Lecture Notes in Math., Vol. 345. MR**0380478****3.**J. W. Calkin,*Two-sided ideals and congruences in the ring of bounded operators in Hilbert space*, Ann. of Math. (2)**42**(1941), 839–873. MR**0005790****4.**R. G. Douglas,*Banach algebra techniques in operator theory*, Academic Press, New York, 1972.**50:14335****5.**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**0322534****6.**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**, 10.1090/S0002-9939-1982-0643740-0**7.**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**, 10.1090/S0002-9939-1986-0840628-0**8.**L. Grunenfelder, M. Omladic, H. Radjavi, and A. Sourour,*Semigroups generated by similarity orbits*, Semigroup Forum**62**(2001), no. 3, 460–472. MR**1831467**, 10.1007/s002330010029**9.**Paul R. Halmos and Shizuo Kakutani,*Products of symmetries*, Bull. Amer. Math. Soc.**64**(1958), 77–78. MR**0100225**, 10.1090/S0002-9904-1958-10156-1**10.**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**, 10.4153/CMB-1995-033-9**11.**H. Radjavi,*The group generated by involutions*, Proc. Roy. Irish Acad. Sect. A**81**(1981), no. 1, 9–12. MR**635572****12.**Heydar Radjavi and Peter Rosenthal,*Invariant subspaces*, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77. MR**0367682****13.**Pei Yuan Wu,*The operator factorization problems*, Linear Algebra Appl.**117**(1989), 35–63. MR**993030**, 10.1016/0024-3795(89)90546-6

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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

DOI:
https://doi.org/10.1090/S0002-9939-03-06910-7

Keywords:
Semigroups,
conjugation-invariant

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

Article copyright:
© Copyright 2003
American Mathematical Society