# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## The commutant of analytic Toeplitz operatorsHTML articles powered by AMS MathViewer

by James A. Deddens and Tin Kin Wong
Trans. Amer. Math. Soc. 184 (1973), 261-273 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 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 $\{ {T_\phi }\} β = \{ {T_\chi }\} β \cap \{ {T_F}\} β$. 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
Similar Articles
• Retrieve articles in Transactions of the American Mathematical Society with MSC: 47B35
• Retrieve articles in all journals with MSC: 47B35