Powers and roots of Toeplitz operators

Louhichi, Issam

doi:10.1090/S0002-9939-06-08626-6

Powers and roots of Toeplitz operators
HTML articles powered by AMS MathViewer

by Issam Louhichi PDF

Proc. Amer. Math. Soc. 135 (2007), 1465-1475 Request permission

Abstract:

We study the commutativity of two Toeplitz operators whose symbols are quasihomogeneous functions. We give a relationship between this commutativity and the roots (or powers) of the Toeplitz operators. We use this to characterize Toeplitz operators with symbols in $L^{\infty }(\mathbb {D})$ which commute with Toeplitz operators whose symbols are of the form $e^{ip\theta }r^m$.

References

Patrick Ahern and Željko Čučković, A theorem of Brown-Halmos type for Bergman space Toeplitz operators, J. Funct. Anal. 187 (2001), no. 1, 200–210. MR 1867348, DOI 10.1006/jfan.2001.3811
Sheldon Axler and Željko Čučković, Commuting Toeplitz operators with harmonic symbols, Integral Equations Operator Theory 14 (1991), no. 1, 1–12. MR 1079815, DOI 10.1007/BF01194925
Sheldon Axler, Željko Čučković, and N. V. Rao, Commutants of analytic Toeplitz operators on the Bergman space, Proc. Amer. Math. Soc. 128 (2000), no. 7, 1951–1953. MR 1694299, DOI 10.1090/S0002-9939-99-05436-2
Arlen Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/64), 89–102. MR 160136, DOI 10.1007/978-1-4613-8208-9_{1}9
eljko C̆u ković and N. V. Rao, Mellin transform, monomial symbols, and commuting Toeplitz operators, J. Funct. Anal. 154 (1998), no. 1, 195–214. MR 1616532, DOI 10.1006/jfan.1997.3204
I. Louhichi, E. Strouse, L. Zakariasy, Products of Toeplitz operators on the Bergman space, Integral Equations and Operator Theory. Published on line: 1 October 2005.
Issam Louhichi and Lova Zakariasy, On Toeplitz operators with quasihomogeneous symbols, Arch. Math. (Basel) 85 (2005), no. 3, 248–257. MR 2172383, DOI 10.1007/s00013-005-1198-0
Reinhold Remmert, Classical topics in complex function theory, Graduate Texts in Mathematics, vol. 172, Springer-Verlag, New York, 1998. Translated from the German by Leslie Kay. MR 1483074, DOI 10.1007/978-1-4757-2956-6