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

Abstract:

Let $\gamma (S)$ denote the reduced minimum modulus of a linear operator $S$ acting in a complex Banach space $X$, and let $I$ denote the identity on $X$. In this paper it is shown that for a (not necessarily bounded) Fredholm operator $T$ acting in $X$, the limit ${\lim _{n \to \infty }}\gamma {({T^n})^{1/n}}$ exists and is equal to the supremum of all positive numbers $\delta$ such that the dimension of the null space and the codimension of the range of $T - \lambda I$ are constant on $0 < |\lambda | < \delta$.