The Commutant of an Abstract Backward Shift

http://dx.doi.org/10.4153/CMB-2000-003-9
Canad. Math. Bull. 43(2000), 21-24

  Published:2000-03-01
 Printed: Mar 2000

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.