Abstract. Assume that $T$ is a bounded linear operator on a Banach space $X$. Let $W$ be a closed $T$-invariant subspace of $X.$ In this paper, the relationships between the spectral and Fredholm properties of $T$ and those of the pair of operators $T_{W}$ and $\widehat{T_{W}}$ are studied ($T_{W}$ is the restriction of $T$ to $W$ and $\widehat{T_{W}}$ is the operator determined by $T$ on $X/W).$ These results are applied to operators with infinite ascent or infinite strong descent.

AMS Subject Classification (1991): 47A05, 47A10, 47A53

Received September 21, 2005. (Registered under 5964/2009.)