Abstract
We examine the spectra of the multiplier algebras of the Dirichlet space and of the generalized Dirichlet spaces (spaces between the Hardy space and the Dirichlet space). For the generalized Dirichlet spaces we show that the spectrum contains nontrivial analytic disks. For the Dirichlet space we are able to show that the spectrum contains nontrivial Gleason parts.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Agler, J.; McCarthy, J. Pick interpolation and Hilbert function spaces. American Mathematical Society, Providence, RI, 2002.
Arcozzi, N. Rochberg, R. Topics in dyadic Dirichlet spaces. New York J. Math. 10 (2004), 45–67
Arcozzi, N.; Rochberg, R.; Sawyer, E.; Wick, B. D. Distance functions for reproducing kernel Hilbert spaces. Function spaces in modern analysis, 25–53, Contemp. Math., 547, Amer. Math. Soc., Providence, RI, 2011.
Bøe, B. An interpolation theorem for Hilbert spaces with Nevanlinna-Pick kernel. Proc. Amer. Math. Soc. 133 (2005), no. 7, 2077–2081
Carleson, L On a Class of Meromorphic Functions and Its Associated Exceptional Sets. Thesis, University of Uppsala, 1950.
Chow, K. N.; Protas, D. The bounded, analytic, Dirichlet finite functions and their fibers. Arch. Math. (Basel) 33 (1979/80), no. 6, 575–582.
Chow, K. N.; Protas, D The maximal ideal space of bounded, analytic, Dirichlet finite functions. Arch. Math. (Basel) 31 (1978/79), no. 3, 298–301.
Cole, B. J.; Gamelin, T. W.; Johnson, W. B. Analytic disks in fibers over the unit ball of a Banach space. Michigan Math. J. 39 (1992), no. 3, 551–569.
Duren, P,; Weir, R. The pseudohyperbolic metric and Bergman spaces in the ball. Trans. Amer. Math. Soc. 359 (2007), no. 1, 63–76.
Gamelin, T. W. On analytic disks in the spectrum of a uniform algebra. Arch. Math. (Basel) 52 (1989), no. 3, 269–274.
Garnett, J. Bounded analytic functions. Academic Press, Inc. 1981.
Gleason, A. Function Algebras, Seminars on Analytic Functions Vol II, IAS, Princeton, 1957
Hoffman, K. Banach spaces of analytic functions. Prentice-Hall, Inc., Englewood Cliffs, N. J. 1962
Hoffman, K. Bounded analytic functions and Gleason parts. Ann. of Math. (2) 86 1967 74–111.
Izuchi, K. J.; Izuchi, Y. Factorization of Blaschke products and ideal theory in H ∞. J. Funct. Anal. 260 (2011), no. 7, 2086–2147
Rochberg, R.; Wu, Z. J. A new characterization of Dirichlet type spaces and applications. Illinois J. Math. 37 (1993), no. 1, 101–122.
Ross, W. The classical Dirichlet space. Recent advances in operator-related function theory, 171–197, Contemp. Math., 393, AMS Providence, RI, 2006.
Sawyer, E. Function theory: interpolation and corona problems. Fields Institute Monographs, 25. AMS Providence, RI; 2009.
Schark, I. J. Maximal ideals in an algebra of bounded analytic functions. J. Math. Mech. 10 1961 735–746.
Verbitskiĭ, I. È. Multipliers in spaces with “fractional” norms, and inner functions. Sibirsk. Mat. Zh. 26 (1985), no. 2, 51–72, 221.
Wu, Z. J. Carleson measures and multipliers for Dirichlet spaces. J. Funct. Anal. 169 (1999), no. 1, 148–163.
Acknowledgements
This work was supported by the National Science Foundation under Grant No. 1001488.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this chapter
Cite this chapter
Rochberg, R. (2014). Structure in the Spectra of Some Multiplier Algebras. In: Douglas, R., Krantz, S., Sawyer, E., Treil, S., Wick, B. (eds) The Corona Problem. Fields Institute Communications, vol 72. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1255-1_9
Download citation
DOI: https://doi.org/10.1007/978-1-4939-1255-1_9
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-1254-4
Online ISBN: 978-1-4939-1255-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)