On the spectral character

of Toeplitz operators on planar regions

Author:
Gholamreza Akbari Estahbanati

Journal:
Proc. Amer. Math. Soc. **124** (1996), 2737-2744

MSC (1991):
Primary 47B35

DOI:
https://doi.org/10.1090/S0002-9939-96-03323-0

MathSciNet review:
1326992

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Abstract | References | Similar Articles | Additional Information

Abstract: Self-adjoint Toeplitz operators on multiply connected planar regions are investigated using theta functions on the double. An explicit resolvent form for self-adjoint Toeplitz operators on a Hardy space associated with any representing measure on a -holed planar region is given via reproducing kernels in terms of theta functions on . This resolvent formula is a generalization of an analogous formula obtained by K. F. Clancey (1991) for the case of harmonic measure. Applications of this resolvent form to the spectral type of the self-adjoint Toeplitz operators are described.

**[1]**G. Akbari Estahbanati,*Riemann surfaces and Toeplitz operators on multiply connected planar regions*, Dissertation, University of Georgia, 1993.**[2]**J. A. Ball and K. F. Clancey,*Reproducing kernels for Hardy spaces on multiply connected domains*.**[3]**K. F. Clancey,*Toeplitz operators on multiply connected domains and theta functions*, Operator Theory: Adv. and Appl.**35**(1988), 311--355. MR**91f:47038****[4]**------,*Representing measures on multiply connected planar domains*, Illinois J. Math., vol. 35**2**(1991), 286--311. MR**92e:46110****[5]**------,*On the spectral character of Toeplitz operators on multiply connected domains*, Trans. Amer. Math. Soc., vol. 323**2**(1991), 897--910. MR**91f:47039****[6]**W. F. Donoghue, Jr.,*On the perturbation of spectra*, Comm. Pure Appl. Math.,**18**(1965), 559--579. MR**32:8171****[7]**J. D. Fay,*Theta Functions on Riemann Surfaces, Lecture Notes in Mathematics No. 352*, Springer-Verlag, New York, 1973. MR**49:569****[8]**D. Mumford,*Tata Lectures on Theta. I, II*, Birkhäuser Verlag, Basel, 1983. MR**86b:14017,****85h:**14026**[9]**J. D. Pincus and J. Xia,*Symmetric and self-adjoint Toeplitz operators on multiply connected planar domains*, J. Funct. Anal.**59**(1984), 397--444. MR**87i:47038****[10]**M. Rosenblum,*A concrete spectral theory for self-adjoint Toeplitz operators*, Amer. J. Math.**87**(1965), 709--718. MR**31:6127****[11]**E. I. Zverovich,*Boundary value problems in the theory of analytic functions in Hölder classes on Riemann surfaces*, Russian Mathematical Surveys**26**(1971), 117--192. MR**53:13593**

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Additional Information

**Gholamreza Akbari Estahbanati**

Affiliation:
Department of Mathematics and Computer Science, North Georgia College, Dahlonega, Georgia 30597

Email:
akbari@nugget.ngc.peachnet.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03323-0

Keywords:
Riemann surface,
double,
representing measure,
theta function,
Hardy space,
Toeplitz operator

Received by editor(s):
September 13, 1994

Received by editor(s) in revised form:
January 30, 1995, and March 3, 1995

Communicated by:
Albert Baernstein II

Article copyright:
© Copyright 1996
American Mathematical Society