## Arkiv för Matematik

• Ark. Mat.
• Volume 38, Number 1 (2000), 97-110.

### Indices, characteristic numbers and essential commutants of Toeplitz operators

Kunyu Guo

#### Abstract

For an essentially normal operator T, it is shown that there exists a unilateral shift of multiplicity m in C*(T) if and only if γ(T)≠0 and γ(T)/m. As application, we prove that the essential commutant of a unilateral shift and that of a bilateral shift are not isomorphic as C*-algebras. Finally, we construct a natural C*-algebra ε + ε* on the Bergman space L ${}_{a}^{2}$ (Bn), and show that its essential commutant is generated by Toeplitz operators with symmetric continuous symbols and all compact operators.

#### Note

Supported by NSFC and Laboratory of Mathematics for Nonlinear Science at Fudan University.

#### Article information

Source
Ark. Mat., Volume 38, Number 1 (2000), 97-110.

Dates
Revised: 15 December 1998
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485898669

Digital Object Identifier
doi:10.1007/BF02384493

Mathematical Reviews number (MathSciNet)
MR1749361

Zentralblatt MATH identifier
1021.47018

Rights

#### Citation

Guo, Kunyu. Indices, characteristic numbers and essential commutants of Toeplitz operators. Ark. Mat. 38 (2000), no. 1, 97--110. doi:10.1007/BF02384493. https://projecteuclid.org/euclid.afm/1485898669

#### References

• Arveson, W., Notes on the extensions of C*-algebras, Duke Math. J. 44 (1977), 329–354.
• Barria, J. and Halmos, P. R., Asymptotic Toeplitz operators, Trans. Amer. Math. Soc. 273 (1982), 621–630.
• Bekolle, D., Berger, C. A., Coburn, L. A. and Zhu, K. H., BMO in the Bergman metric on bounded symmetric domains, J. Funct. Anal. 93 (1990), 921–953.
• Brown, L. G., Douglas, R. G. and Fillmore, P. A., Unitary equivalence modulo the compact operators and extensions of C*-algebras, in Proceedings of a Conference on Operator Theory (Fillmore, P. A., ed.), Lecture Notes in Math. 345, pp. 58–128, Springer-Verlag, Berlin-Heidelberg, 1973.
• Coburn, L. A., Singular integral operators and Toeplitz operators on odd spheres, Indiana Univ. Math. J. 23 (1973/74), 433–439.
• Douglas, R. G., Banach Algebra Techniques in Operator Theory, Academic Press, New York, 1972.
• Englis, M., Density of algebras generated by Toeplitz operators on Bergman spaces, Ark. Mat. 30 (1992), 227–240.
• Englis, M., Toeplitz operators on Cartan domains essentially commute with a bilateral shift, Proc. Amer. Math. Soc. 117 (1993), 365–368.
• Gohberg, I. C. and Krein, M. G., Systems of integral equations on a half-line with kernels depending on the difference of arguments, Uspekhi Mat. Nauk 13:2 (1958), 3–72 (Russian). English transl.: Amer. Math. Soc. Transl. 14 (1960), 217–287.
• Liu, J., Zhang, Y. and Guo, K., A new C*-algebra and its essential commutant, Chinese Sci. Bull. 44 (1999), 204–207.
• McDonald, G. Fredholm properties of a class of Toeplitz operators on the ball, Indiana Univ. Math. J. 26 (1977), 567–576.
• Power, S. C., C*-modules and an odd-even decomposition for C*-algebras, Bull. London Math. Soc. 8 (1976), 268–272.
• Salinas, N., The ∂-formalism and the C*-algebra of the Bergman n-tuple, J. Operator Theory 22 (1989), 325–343.
• Zhu, K. H., Duality and Hankel operators on the Bergman spaces of bounded symmetric domains, J. Funct. Anal. 81 (1988), 260–278.