  While in \cite{HB} we studied classes of Fredholm-type operators defined by
the homomorphism $\Pi$ from $L(X)$ onto the Calkin algebra $\mathcal{C}(X)$,
$X$ being a Banach space, we study in this paper two classes of Fredholm-type
operators defined by the homomorphism $\pi$ from $L(X)$ onto the algebra
$\mathcal{C}_0(X)= L(X)/F_0(X),$ where $F_0(X)$ is the ideal of finite rank
operators in $L(X).$ Then we define an index for Fredholm-type operators and we
show that this new index satisfies similar properties as the usual Fredholm
index.
