The asymptotic behaviour of the reduced minimum modulus of a Fredholm operator

Let $ \gamma (S)$ acting in a complex Banach space $ X$ denote the identity on $ X$ acting in $ X$ $ {\lim _{n \to \infty }}\gamma {({T^n})^{1/n}}$ such that the dimension of the null space and the codimension of the range of $ T - \lambda I$ <IMG WIDTH="101" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img11.gif" ALT="$ 0 &lt; \vert\lambda \vert &lt; \delta $">.