The commutant of analytic Toeplitz operators
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 by James A. Deddens and Tin Kin Wong PDF
 Trans. Amer. Math. Soc. 184 (1973), 261273 Request permission
Abstract:
In this paper we study the commutant of an analytic Toeplitz operator. For $\phi \;\;{H^\infty }$, let $\phi = \chi F$ be its innerouter 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 $\{ {T_\phi }\} β = \{ {T_\chi }\} β \cap \{ {T_F}\} β$. 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 $\{ {T_\phi }\} β = \{ {T_\chi }\} β \cap \{ {T_F}\} β$, another is that if $\phi \;\epsilon {H^\infty }$ is univalent then $\{ {T_\phi }\} β = \{ {T_z}\} β$. We are also able to prove that if the inner factor of $\phi$ is $\chi (z) = {z^n},n \geq 1$, then $\{ {T_\phi }\} β = \{ {T_{{z^s}}}\} β$ 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.References

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Additional Information
 © Copyright 1973 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 184 (1973), 261273
 MSC: Primary 47B35
 DOI: https://doi.org/10.1090/S00029947197303244670
 MathSciNet review: 0324467