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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

The commutant of analytic Toeplitz operators


Authors: James A. Deddens and Tin Kin Wong
Journal: Trans. Amer. Math. Soc. 184 (1973), 261-273
MSC: Primary 47B35
MathSciNet review: 0324467
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Abstract: In this paper we study the commutant of an analytic Toeplitz operator. For $ \phi \;\;{H^\infty }$, let $ \phi = \chi F$ be its inner-outer factorization. Our main result is that if there exists $ \lambda \;\epsilon \;{\text{C}}$ such that X factors as $ \chi = {\chi _1}{\chi _2} \cdots {\chi _n}$, each $ {\chi _i}$ an inner function, and if $ F - \lambda $ is divisible by each $ {\chi _i}$, then $ \{ {T_\phi }\} ' = \{ {T_\chi }\} ' \cap \{ {T_F}\} '$. The key step in the proof is Lemma 2, which is a curious result about nilpotent operators. One corollary of our main result is that if $ \chi (z) = {z^n},n \geq 1$, then $ \{ {T_\phi }\} ' = \{ {T_\chi }\} ' \cap \{ {T_F}\} '$, another is that if $ \phi \;\epsilon {H^\infty }$ is univalent then $ \{ {T_\phi }\} ' = \{ {T_z}\} '$. We are also able to prove that if the inner factor of $ \phi $ is $ \chi (z) = {z^n},n \geq 1$, then $ \{ {T_\phi }\} ' = \{ {T_{{z^s}}}\} '$ where s is a positive integer maximal with respect to the property that $ {z^n}$ and $ F(z)$ are both functions of $ {z^s}$. We conclude by raising six questions.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0324467-0
PII: S 0002-9947(1973)0324467-0
Keywords: Analytic function, inner and outer functions, $ {H^\infty }$, $ {H^2}$, analytic Toeplitz operator, pure is ometry, commutant
Article copyright: © Copyright 1973 American Mathematical Society