On the toeplitzness of composition operators

Abstract

Composition operators on H ² cannot–except trivially–be Toeplitz, or even ‘Toeplitz plus compact’. However there are natural ways in which they can be ‘asymptotically Toeplitz’. We show here that the study of such phenomena leads to surprising results and interesting open problems.

†Dedicated to Professor Peter L. Duren on the occasion of his 70th birthday.

Keywords:

• Composition operator
• Asymptotic Toeplitz operator

Keywords:

• AMS 2000 Mathematics Subject Classification
• 47B33
• 47B35

Acknowledgements

We wish to thank Ruben Martinez-Avandaño for useful discussions during the preliminary stages of this work, and for introducing us to Feintuch's work 7. We also thank Paul Bourdon and Valentin Matache for a number of suggestions and corrections which improved the manuscript, and Wayne Smith for useful discussions during the project's preliminary phase. Research supported in part by NSF grants DMS 9622396 and DMS 0100502.

Notes

†Dedicated to Professor Peter L. Duren on the occasion of his 70th birthday.

†See, e.g., 15. If ω has C ^∞ boundry then the extension is C ^∞ on ∂ U; this is the classical theorem of Panilevé–form a nice exposition see 16.