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# The Commutant of an Abstract Backward Shift

Published:2000-03-01
Printed: Mar 2000
• Bruce A. Barnes
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## Abstract

A bounded linear operator $T$ on a Banach space $X$ is an abstract backward shift if the nullspace of $T$ is one dimensional, and the union of the null spaces of $T^k$ for all $k \geq 1$ is dense in $X$. In this paper it is shown that the commutant of an abstract backward shift is an integral domain. This result is used to derive properties of operators in the commutant.
 Keywords: backward shift, commutant
 MSC Classifications: 47A99 - None of the above, but in this section