Hankel Operators on Bounded Analytic Functions

Authors:
James Dudziak, T. W. Gamelin and Pamela Gorkin

Journal:
Trans. Amer. Math. Soc. **352** (2000), 363-377

MSC (1991):
Primary 46J15, 47B38; Secondary 30D55, 47B05

DOI:
https://doi.org/10.1090/S0002-9947-99-02178-9

Published electronically:
July 21, 1999

MathSciNet review:
1473437

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a domain in the complex plane and a bounded measurable function on , the generalized Hankel operator on is the operator of multiplication by followed by projection into . Under certain conditions on we show that either is compact or there is an embedded on which is bicontinuous. We characterize those 's for which is compact in the case that is a Behrens roadrunner domain.

**1.**M. Behrens,*The corona conjecture for a class of infinitely connected domains*, Bull. Amer. Math. Soc.**76**(1970), 387-391. MR**41:825****2.**M. Behrens,*The maximal ideal space of algebras of bounded analytic functions on infinitely connected domains*, Trans. Amer. Math. Soc.**161**(1971), 359-380. MR**55:8380****3.**J. Bourgain,*The Dunford-Pettis property for the ball-algebras, the polydisc-algebras and the Sobolev spaces*, Studia Math.**77**(1984), 245-253. MR**85f:46044****4.**J. Bourgain,*New Banach space properties of the disc algebra and*, Acta Math.**152**(1984), 1-48. MR**85j:46091****5.**J. A. Cima, S. Janson and K. Yale,*Completely continuous Hankel operators on and Bourgain algebras*, Proc. Amer. Math. Soc.**105**(1989), 121-125. MR**89g:30065****6.**J. A. Cima, K. Stroethoff and K. Yale,*Bourgain algebras on the unit disk*, Pacific J. Math.**160**(1993), 27-41. MR**94i:46065****7.**J. A. Cima and R. M. Timoney,*The Dunford-Pettis property for certain planar uniform algebras*, Michigan Math. J.**34**(1987), 99-104. MR**88e:46023****8.**B. J. Cole and T. W. Gamelin,*Tight uniform algebras and algebras of analytic functions*, J. Funct. Anal.**46**(1982), 158-220. MR**83h:46065****9.**B. J. Cole and T. W. Gamelin,*Weak-star continuous homomorphisms and a decomposition of orthogonal measures*, Ann. Inst. Fourier (Grenoble)**35**(1985), 149-189. MR**86m:46051****10.**A. M. Davie, T. W. Gamelin, and J. W. Garnett,*Distance estimates and pointwise bounded density*, Trans. Amer. Math. Soc.**175**(1973), 37-68. MR**47:2068****11.**T. W. Gamelin,*Lectures on*, Notas de Matemática, No.21, Universidad Nacional de La Plata, Argentina, 1972.**12.**T. W. Gamelin,*Uniform Algebras*, 2nd edition, Chelsea Press, 1984. MR**53:14137 (1st ed.)****13.**T. W. Gamelin,*Localization of the corona problem*, Pacific J. Math.**34**(1970), 73-81. MR**43:2482****14.**T. W. Gamelin,*Uniform algebras on plane sets*, in*Approximation Theory*, G.G.Lorentz (ed), Academic Press, 1973, 101-149. MR**49:3548****15.**T. W. Gamelin and J. Garnett,*Distinguished homomorphisms and fiber algebras*, Amer. J. Math.**92**(1970), 455-474. MR**46:2434****16.**J. Garnett,*Bounded Analytic Functions*, Academic Press, 1981. MR**83g:30037****17.**P. Gorkin, K. Izuchi and R. Mortini,*Bourgain algebras of Douglas algebras*, Canad. J. Math.**44**(1992), 797-804. MR**94c:46104****18.**P. Gorkin and Z. Zheng, in preparation.**19.**K. Izuchi,*Bourgain algebras of the disk, polydisk, and ball algebras*, Duke Math. J.**66**(1992), 503-519. MR**93f:46082****20.**S. F. Saccone,*Banach space properties of strongly tight uniform algebras*, Studia Math.**114**(1995), 159-180. MR**96d:46068****21.**L. Zalcman,*Bounded analytic functions on domains of infinite connectivity*, Trans. Amer. Math. Soc**144**(1969), 241-269. MR**40:5884**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
46J15,
47B38,
30D55,
47B05

Retrieve articles in all journals with MSC (1991): 46J15, 47B38, 30D55, 47B05

Additional Information

**James Dudziak**

Affiliation:
Lyman Briggs School, Michigan State University, East Lansing, Michigan 48825

Email:
dudziak@pilot.msu.edu

**T. W. Gamelin**

Affiliation:
Department of Mathematics, University of California, Los Angeles, California 90024

Email:
gamelin@math.ucla.edu

**Pamela Gorkin**

Affiliation:
Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837

Email:
pgorkin@bucknell.edu

DOI:
https://doi.org/10.1090/S0002-9947-99-02178-9

Received by editor(s):
May 6, 1997

Published electronically:
July 21, 1999

Article copyright:
© Copyright 1999
American Mathematical Society