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

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

Published electronically:
May 9, 2003

MathSciNet review:
1992861

Full-text PDF Free Access

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.**A. Brown and C. Pearcy,*Structure of commutators of operators*, Ann. Math.**82**(1965), 112-127. MR**31:2612****2.**L. G. Brown, R. G. Douglas and P. A. Fillmore,*Unitary equivalence modulo the compact operators and extensions of -algebras*, pp. 58-128, Lecture Notes in Mathematics, Vol. 345, 1973, Springer-Verlag, Berlin. MR**52:1378****3.**J. W. Calkin,*Two-sided ideals and congruences in the ring of bounded operators in Hilbert space*, Ann. Math. (2)**42**(1941), 839-873. MR**3:208c****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**48:896****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), 516-520. MR**84g:47039****7.**C. K. Fong and A. R. Sourour,*The group generated by unipotent operators*, Proc. Amer. Math. Soc.**97**(1986), 453-458. MR**87m:47099****8.**L. Grunenfelder, M. Omladic, H. Radjavi and A. R. Sourour,*Semigroups generated by similarity orbits*, Semigroup Forum**62**(2001), 460-472. MR**2002g:20107****9.**P. R. Halmos and S. Kakutani,*Products of symmetries*, Bull. Amer. Math. Soc.**64**(1958), 77-78. MR**20:6658****10.**N. C. Phillips,*Every invertible Hilbert space operator is a product of seven positive operators*, Canad. Math. Bull.**38**(1995), 230-236. MR**96h:47044****11.**H. Radjavi,*The group generated by involutions*, Proc. Roy. Irish. Acad., Section A,**81**(1981), 9-12. MR**83c:47006****12.**H. Radjavi and P. Rosenthal,*Invariant subspaces*, Springer-Verlag, Berlin, 1973. MR**51:3924****13.**P. Y. Wu,*The operator factorization problems*, Linear Algebra Appl.**117**(1989), 35-63. MR**90g:47031**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
47D03,
20F38

Retrieve articles in all journals with MSC (2000): 47D03, 20F38

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