Fredholm-type Operators and Index

While in \cite{HB} we studied classes of Fredholm-type operators defined by the homomorphism $\mathrm{\Pi }$ 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 ${\mathcal{C}}_0(X)=L(X)\mathrm{/}{F}_0(X),$ where $F_0(X)$ is the ideal of finite rank operators in $L(X)\mathrm{.}$ Then we define an index for Fredholm-type operators and we show that this new index satisfies similar properties as the usual Fredholm index.