On some properties of Banach operators

Abstract

A mapping α from a normed space X into itself is called a Banach operator if there is a constant k such that 0≤k<1 and ‖α2(x)−α(x)‖≤k‖α(x)−x‖ for all x∈X. In this note we study some properties of Banach operators. Among other results we show that if α is a linear Banach operator on a normed space X, then N(α−1)=N((α−1)2), N(α−1)∩R(α−1)=(0) and if X is finite dimensional then X=N(α−1)⊕R(α−1), where N(α−1) and R(α−1) denote the null space and the range space of (α−1), respectively and 1 is the identity mapping on X. We also obtain some commutativity results for a pair of bounded linear multiplicative Banach operators on normed algebras.