On the reflexivity of operators on function spaces

Seddighi, K.; Yousefi, B.

doi:10.1090/S0002-9939-1992-1104402-5

On the reflexivity of operators on function spaces
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by K. Seddighi and B. Yousefi

Proc. Amer. Math. Soc. 116 (1992), 45-52

DOI: https://doi.org/10.1090/S0002-9939-1992-1104402-5

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Abstract:

Let $\Omega$ be a bounded plane domain. Sufficient conditions are given so that an operator $T$ in the Cowen-Douglas class ${\mathcal {B}_n}(\Omega )$ is reflexive. The operator ${M_z}$ of multiplication by $z$ on a Hilbert space of functions analytic on a finitely connected domain $\Omega$ is shown to be reflexive whenever $\sigma ({M_z}) = \overline \Omega$ is a spectral set.

References

H. Bercovici, C. Foiaş, J. Langsam, and C. Pearcy, (BCP)-operators are reflexive, Michigan Math. J. 29 (1982), no. 3, 371–379. MR 674290
B. Chevreau, C. M. Pearcy, and A. L. Shields, Finitely connected domains $G$, representations of $H^{\infty }(G)$, and invariant subspaces, J. Operator Theory 6 (1981), no. 2, 375–405. MR 643698
M. J. Cowen and R. G. Douglas, Complex geometry and operator theory, Acta Math. 141 (1978), no. 3-4, 187–261. MR 501368, DOI 10.1007/BF02545748
Raúl E. Curto and Norberto Salinas, Generalized Bergman kernels and the Cowen-Douglas theory, Amer. J. Math. 106 (1984), no. 2, 447–488. MR 737780, DOI 10.2307/2374310
James A. Deddens, Every isometry is reflexive, Proc. Amer. Math. Soc. 28 (1971), 509–512. MR 278099, DOI 10.1090/S0002-9939-1971-0278099-7
J. A. Deddens and P. A. Fillmore, Reflexive linear transformations, Linear Algebra Appl. 10 (1975), 89–93. MR 358390, DOI 10.1016/0024-3795(75)90099-3

Uniform algebras

D. Sarason, Invariant subspaces and unstarred operator algebras, Pacific J. Math. 17 (1966), 511–517. MR 192365
Karim Seddighi, Essential spectra of operators in the class ${\cal B}_{n}(\Omega )$, Proc. Amer. Math. Soc. 87 (1983), no. 3, 453–458. MR 684638, DOI 10.1090/S0002-9939-1983-0684638-2
Allen L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. MR 0361899
A. L. Shields and L. J. Wallen, The commutants of certain Hilbert space operators, Indiana Univ. Math. J. 20 (1970/71), 777–788. MR 287352, DOI 10.1512/iumj.1971.20.20062