Abstract
The reflexivity and transitivity of subspaces of Toeplitz operators on the Hardy space over a Jordan region — a simply connected region in the complex plane with analytic Jordan curve as its boundary — is investigated. The dichotomic behavior (either reflexivity or transitivity) of these subspaces is shown. It refers to the similar dichotomic behavior of subspaces of Toeplitz operators on the Hardy space over the unit disc.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. B. Abrahamse, Toeplitz Operators in Multiply Connected Regions, Amer. J. Math., 96 (1974), 261–297.
M. B. Abrahamse and R. G. Douglas, A class of subnormal operators related to multiply-connected domains, Adv. Math., 19 (1976), 106–147.
E. Azoff, On finite rank operators and preannihilators, Memoirs of the Amer. Math. Soc. 64, Amer. Math. Soc., Providence, Rhode Island, 1986.
E. A. Azoff and M. Ptak, A Dichotomy for Linear Spaces of Toeplitz Operators, J. Funct. Anal., 156 (1998), 411–428.
H. Bercovici, C. Foias and C. Pearcy, Dual algebras with applications to invariant subspaces and dilation theory, CBMS Regional Conf. Ser. in Math. 56, Amer. Math. Soc., Providence, R.I., 1985.
S. Brown, B. Chevreau and C. Pearcy, Contractions with rich spectrum have invariant subspaces, J. Operator Theory, 1 (1979), 123–136.
J. B. Conway, A Course in Functional Analysis, Springer, New York, 1990.
J. B. Conway, A Course in Operator Theory, Amer. Math. Soc., 2000.
J. B. Conway, The harmonic functional calculus and hyperreflexivity, Pacific J. Math., 204 (2002), 19–29.
R. G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New York, 1972.
S. Fisher, Function Theory on Planar Domains, A Second Course in Complex Analysis, Wiley, New York, 1983.
J. Kraus and D. Larson, Reflexivity and distance formulae, Proc. Lond. Math. Soc., 53 (1986), 340–356.
W. Młocek and M. Ptak, On the reflexivity of subspaces of Toeplitz operators on the Hardy space on the upper half-plane, Czechoslovak Math. J., 63 (2013), 421–434.
N. K. Nikolski, Operators, Functions, and Systems: an easy reading, Vol. 1: Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs 92, Amer. Math. Soc., 2002.
W. Rudin, Analytic functions of class Hp, Transactions Amer. Math. Soc., 78 (1955), 46–66.
D. Sarason, Invariant subspaces and unstarred operator algebras, Pacific J. Math., 17 (1966), 511–517.
Acknowledgements
The first author was supported by a grant founded by the Rector of the University of Agriculture in Krakow.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Dedicated to the memory of Professor Béla Szőkefalvi-Nagy
Communicated by L. Kérchy
Rights and permissions
About this article
Cite this article
Młocek, W., Ptak, M. On the reflexivity of subspaces of Toeplitz operators in simply connected regions. ActaSci.Math. 80, 275–287 (2014). https://doi.org/10.14232/actasm-012-343-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.14232/actasm-012-343-0