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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Toeplitz operators with PC symbols
on general Carleson Jordan curves
with arbitrary Muckenhoupt weights


Authors: Albrecht Böttcher and Yuri I. Karlovich
Journal: Trans. Amer. Math. Soc. 351 (1999), 3143-3196
MSC (1991): Primary 47B35; Secondary 30E20, 42A50, 45E05, 47D30
Published electronically: March 29, 1999
MathSciNet review: 1650069
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Abstract: We describe the spectra and essential spectra of Toeplitz operators with piecewise continuous symbols on the Hardy space $H^p(\Gamma,\omega)$ in case $1<p<\infty $, $\Gamma$ is a Carleson Jordan curve and $\omega$ is a Muckenhoupt weight in $A_p(\Gamma)$. Classical results tell us that the essential spectrum of the operator is obtained from the essential range of the symbol by filling in line segments or circular arcs between the endpoints of the jumps if both the curve $\Gamma$ and the weight are sufficiently nice. Only recently it was discovered by Spitkovsky that these line segments or circular arcs metamorphose into horns if the curve $\Gamma$ is nice and $\omega$ is an arbitrary Muckenhoupt weight, while the authors observed that certain special so-called logarithmic leaves emerge in the case of arbitrary Carleson curves with nice weights. In this paper we show that for general Carleson curves and general Muckenhoupt weights the sets in question are logarithmic leaves with a halo, and we present final results concerning the shape of the halo.


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Additional Information

Albrecht Böttcher
Affiliation: Faculty of Mathematics, Tech. Univ. Chemnitz-Zwickau, D-09107 Chemnitz, Germany
Email: aboettch@mathematik.tu-chemnitz.de

Yuri I. Karlovich
Affiliation: Faculty of Mathematics, Tech. Univ. Chemnitz-Zwickau, D-09107 Chemnitz, Germany
Address at time of publication: Ukrainian Academy of Sciences, Marine Hydrophysical Institute, Hydroacoustic Department, Preobrazhenskaya Street 3, 270 100 Odessa, Ukraine
Email: karlik@paco.net

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02441-1
PII: S 0002-9947(99)02441-1
Keywords: Carleson condition, Ahlfors--David curve, Muckenhoupt condition, submultiplicative function, Toeplitz operator, singular integral operator, Fredholm operator
Received by editor(s): September 28, 1995
Received by editor(s) in revised form: December 15, 1996
Published electronically: March 29, 1999
Additional Notes: The first author was supported by the Alfried Krupp Förderpreis für junge Hochschullehrer. Both authors were supported by NATO Collaborative Research Grant 950332.
Article copyright: © Copyright 1999 American Mathematical Society