Upper semi-Fredholm and Kato spectrum of an $\alpha$-times integrated semigroup

Abstract

Let $(T(t))_{t\geq 0}$ be an $\alpha$-times integrated semigroup with generator $A$ on a Banach space $X$. In this paper, we show that the spectral mapping theorem holds for upper semi-Fredholm spectrum and we will give some consequences of this result. We also give an application on the Schrödinger operator.