On Some Properties of Banach Operators

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. 2000 Mathematics Subject Classification. 46C05, 47A10, 47A50, 47H10.