Fredholm Equation in Smooth Banach Spaces and Its Applications

The object of this paper is to consider the Fredholm equation (i.e., x-ABx=yo) for smooth Banach spaces. In particular, we will prove that the classical assumption of compactness of AB is redundant in some circumstances. In this paper, we show that Coburn's theorem holds for another classes of generally of nonnormal operators. Moreover, as a corollary, we find the distance from some operator to compact operators.