Abstract. Let $T$ be a bounded linear operator on a separable infinite-dimensional Banach space $X$, and let $N(T)$ denote the nullspace of $T$. We say that $T$ is an abstract backward shift of multiplicity $m$, where $1 \leq m < \infty $, if (1) $\dim N(T^n)/N(T^{n-1}) =m$ for all $n \geq1$ and (2) $\cup_{n=1}^\infty N(T^n)$ is dense in $X$. We characterize the commutant of such an operator, and use our result to determine sufficient conditions for the operator to be irreducible.

AMS Subject Classification (1991): 47A99, 47B37

Keyword(s): backward shift, commutant

Received February 13, 2003, and in revised form 26, 2003. (Registered under 5817/2009.)