Weakly compact composition operators between algebras of bounded analytic functions

Authors:
Pablo Galindo, Mikael Lindström and Ray Ryan

Journal:
Proc. Amer. Math. Soc. **128** (2000), 149-155

MSC (1991):
Primary 46J15; Secondary 46E15, 46G20

DOI:
https://doi.org/10.1090/S0002-9939-99-05196-5

Published electronically:
May 27, 1999

MathSciNet review:
1646307

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a characterization (up to the approximation property) of weakly compact composition operators in terms of their inducing analytic maps .

**[AAD]**R. Alencar, R.M. Aron and S. Dineen,*A reflexive space of holomorphic functions in infinitely many variables*, Proc. Amer. Math. Soc.**90**(1984), 407-411. MR**85b:46050****[ACG]**R.M. Aron, B.J. Cole and T.W. Gamelin,*Spectra of algebras of analytic functions on a Banach space*, J. reine angew. Math.**415**(1991), 51-93. MR**92f:46056****[AF]**R. Alencar and K. Floret,*Weak-strong continuity of multilinear mappings and the Pelczynski -Pitt theorem*, J. of Math. Anal. and Appl.**206 (2)**(1997), 532-546. MR**98h:46045****[AGL]**R. Aron, P. Galindo and M. Lindström,*Compact homomorphisms between algebras of analytic functions*, Studia Math.**123 (3)**(1997), 235-247. MR**98h:46053****[Di]**J. Diestel,*Sequences and series in Banach spaces*, Springer, 1983. MR**85j:46020****[D]**S. Dineen,*Complex analysis in locally convex spaces*, North Holland, 1981. MR**84b:46050****[D1]**S. Dineen,*A Dvoretzky theorem for polynomials*, Proc. Amer. Math. Soc.**123**(1995), 2817-2821. MR**95k:46021****[GG]**M. González and J. Gutiérrez,*Gantmacher type theorems for holomorphic mappings*, Math. Nachr.**186**(1997), 131-145. CMP**97:16****[GGM]**P. Galindo, D. Garcia and M. Maestre,*Entire functions of bounded type on Fréchet spaces*, Math. Nachr.**161**(1993), 185-198. MR**94k:46086****[GL]**P. Galindo and M. Lindström,*Gleason parts and weakly compact homomorphisms between uniform Banach algebras*, Monatshefte für Math.**[G]**J. Garnett,*Bounded analytic functions*, Academic Press, 1981. MR**83g:30037****[HHZ]**P. Habala, P. Hajek and V. Zizler,*Introduction to Banach spaces*, Charles University Prague, 1996.**[JaP]**J. A. Jaramillo and A. Prieto,*The weak-polynomial convergence in a Banach space*, Proc. Amer. Math. Soc.**118**(1993), 463-468.MR**93g:46019****[M1]**J. Mujica,*Linearization of bounded holomorphic mappings on Banach spaces*, Trans. Amer. Math. Soc.**324**(1991), 867-887. MR**91h:46088****[M2]**J. Mujica,*Complex analysis in Banach spaces*, North Holland, 1986. MR**88d:46084****[M3]**J. Mujica,*Complex homomorphisms on the algebras of holomorphic functions on Fréchet spaces*, Math. Ann.**241**(1979), 73-82. MR**80j:46074****[N]**K. Ng,*On a theorem of Dixmier*, Math. Scand.**29**(1971), 279-280. MR**49:3504****[R]**R. Ryan,*Applications of topological tensor products in infinite dimensional holomorphy*, Ph. D. Thesis, Trinity College, Dublin (1980).**[R1]**R. Ryan,*Dunford-Pettis properties*, Bull. Acad. Polon. Sci. Ser. Sci. Math**27**(1979), 373-379. MR**80m:46018****[U]**A. Ülger,*Some results about the spectrum of commutative Banach algebras under the weak topology and applications*, Monatshefte für Math.**121**(1996), 353-379.MR**98a:46058****[V]**M. Valdivia,*Banach spaces of polynomials without copies of*, Proc. Amer. Math. Soc.**123**(1995), 3143-3150. MR**95m:46070**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
46J15,
46E15,
46G20

Retrieve articles in all journals with MSC (1991): 46J15, 46E15, 46G20

Additional Information

**Pablo Galindo**

Affiliation:
Departamento de Análisis Matemático, Universidad de Valencia, 46100 Burjasot, Valencia, Spain

Email:
galindo@uv.es

**Mikael Lindström**

Affiliation:
Department of Mathematics, Åbo Akademi University, FIN-20500 Åbo, Finland

Email:
mikael.lindstrom@abo.fi

**Ray Ryan**

Affiliation:
Department of Mathematics, University College Galway, Galway, Ireland

Email:
ray.ryan@ucg.ie

DOI:
https://doi.org/10.1090/S0002-9939-99-05196-5

Keywords:
Weakly compact operator,
composition operator,
bounded analytic functions on the open unit ball

Received by editor(s):
March 10, 1998

Published electronically:
May 27, 1999

Additional Notes:
Research of the first author was partially supported by DGICYT(Spain) pr. 91-0326.

Communicated by:
Steven R. Bell

Article copyright:
© Copyright 1999
American Mathematical Society