A matricial corona theorem

Authors:
Tavan Trent and Xinjun Zhang

Journal:
Proc. Amer. Math. Soc. **134** (2006), 2549-2558

MSC (2000):
Primary 32A65, 46J20

Published electronically:
April 7, 2006

MathSciNet review:
2213732

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that a usual corona-type theorem on a space of functions automatically extends to a matrix version.

**[1]**G. Birkhoff and S. MacLane,*Algebra*, MacMillan, Toronto, 1971.**[2]**Lennart Carleson,*Interpolations by bounded analytic functions and the corona problem*, Ann. of Math. (2)**76**(1962), 547–559. MR**0141789****[3]**Stephen D. Fisher,*Function theory on planar domains*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1983. A second course in complex analysis; A Wiley-Interscience Publication. MR**694693****[4]**Frank Forelli,*Bounded holomorphic functions and projections*, Illinois J. Math.**10**(1966), 367–380. MR**0193534****[5]**Paul A. Fuhrmann,*On the corona theorem and its application to spectral problems in Hilbert space*, Trans. Amer. Math. Soc.**132**(1968), 55–66. MR**0222701**, 10.1090/S0002-9947-1968-0222701-7**[6]**Artur Nicolau,*The corona property for bounded analytic functions in some Besov spaces*, Proc. Amer. Math. Soc.**110**(1990), no. 1, 135–140. MR**1017007**, 10.1090/S0002-9939-1990-1017007-X**[7]**N. K. Nikol′skiĭ,*Treatise on the shift operator*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin, 1986. Spectral function theory; With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller; Translated from the Russian by Jaak Peetre. MR**827223****[8]**Marvin Rosenblum,*A corona theorem for countably many functions*, Integral Equations Operator Theory**3**(1980), no. 1, 125–137. MR**570865**, 10.1007/BF01682874**[9]**E. L. Stout,*Bounded holomorphic functions on finite Reimann surfaces*, Trans. Amer. Math. Soc.**120**(1965), 255–285. MR**0183882**, 10.1090/S0002-9947-1965-0183882-4**[10]**V. A. Tolokonnikov,*Estimates in Carleson’s corona theorem and finitely generated ideals of the algebra 𝐻^{∞}*, Funktsional. Anal. i Prilozhen.**14**(1980), no. 4, 85–86 (Russian). MR**595742****[11]**S. R. Treil′,*Angles between co-invariant subspaces, and the operator corona problem. The Szőkefalvi-Nagy problem*, Dokl. Akad. Nauk SSSR**302**(1988), no. 5, 1063–1068 (Russian); English transl., Soviet Math. Dokl.**38**(1989), no. 2, 394–399. MR**981054****[12]**Tavan T. Trent,*A corona theorem for multipliers on Dirichlet space*, Integral Equations Operator Theory**49**(2004), no. 1, 123–139. MR**2057771**, 10.1007/s00020-002-1196-6**[13]**Tavan T. Trent,*A new estimate for the vector valued corona problem*, J. Funct. Anal.**189**(2002), no. 1, 267–282. MR**1887635**, 10.1006/jfan.2001.3842**[14]**-,*An -corona theorem on the bidisk for infinitely many functions*, submitted.**[15]**X. Zhang,*A matrix version of corona theorem for algebras of functions on reproducing kernel Hilbert spaces*, Ph.D. dissertation, The University of Alabama, Tuscaloosa, AL, August 2004.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
32A65,
46J20

Retrieve articles in all journals with MSC (2000): 32A65, 46J20

Additional Information

**Tavan Trent**

Affiliation:
Department of Mathematics, The University of Alabama, Box 870350, Tuscaloosa, Alabama 35487-0350

Email:
ttrent@gp.as.ua.edu

**Xinjun Zhang**

Affiliation:
Department of Mathematics, The University of Alabama, Box 870350, Tuscaloosa, Alabama 35487-0350

Email:
zhang010@bama.ua.edu

DOI:
https://doi.org/10.1090/S0002-9939-06-08172-X

Keywords:
Matrix corona theorem

Received by editor(s):
September 8, 2004

Received by editor(s) in revised form:
January 13, 2005

Published electronically:
April 7, 2006

Additional Notes:
This work was partially supported by NSF Grant DMS-0400307.

Communicated by:
Joseph A. Ball

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.