Abstract. Let $T$ be a bounded linear operator on a Banach space $X$. Let $\bf N$ be the generalized null space of $T$ (see the terminology in the first part of the Introduction). If $\bf N$ is dense in $X$, then we call $T$ a general backward shift (a GBS). We show that when $T$ is a GBS, then $T$ has many spectral and Fredholm properties in common with classical weighted backward shifts on $l^2.$ A similar study is made of a related dual concept involving general shifts. Important examples are given of bounded linear operators to which these results apply (see the last section).

AMS Subject Classification (1991): 47A05, 47A10

Received June 16, 2004. (Registered under 5840/2009.)