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Intersections of commutants of analytic Toeplitz operators


Author: James E. Thomson
Journal: Proc. Amer. Math. Soc. 52 (1975), 305-310
MSC: Primary 47B35; Secondary 30A78
DOI: https://doi.org/10.1090/S0002-9939-1975-0399927-4
MathSciNet review: 0399927
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Abstract: In this paper we study the intersection of commutants of analytic Toeplitz operators. Our main result is that if $ \phi $ is a finite Blaschke product and $ \Psi \epsilon {H^\infty }$, then $ \{ {T_\phi }\} ' \cap \{ {T_\Psi }\} ' = \{ {T_I}\} '$ where $ I$ is a finite Blaschke product and $ \phi $ and $ \Psi $ are functions of $ I$. The key step is a function-theoretic theorem describing the relationship between a finite Blaschke product and any $ {H^\infty }$ function.


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  • [1] A. Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/64), 89-102. MR 28 #3350; 30, p. 1205. MR 0160136 (28:3350)
  • [2] J. Deddens and T. K. Wong, The commutant of analytic Toeplitz operators, Trans. Amer. Math. Soc. 184 (1973), 261-273. MR 0324467 (48:2819)
  • [3] R. G. Douglas, Banach algebra techniques in operator theory, Academic Press, New York, 1972. MR 0361893 (50:14335)
  • [4] P. L. Duren, Theory of $ {H^p}$-spaces, Pure and Appl. Math., vol. 38, Academic Press, New York, 1970. MR 42 #3552. MR 0268655 (42:3552)
  • [5] E. Nordgren, Reducing subspaces of analytic Toeplitz operators, Duke Math. J. 34 (1967), 175-181. MR 35 #7155. MR 0216321 (35:7155)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0399927-4
Keywords: Analytic function, inner function, $ {H^\infty }$, $ {H^2}$, analytic Toeplitz operator, commutant
Article copyright: © Copyright 1975 American Mathematical Society

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