International Journal of Mathematics and Mathematical Sciences
Volume 27 (2001), Issue 3, Pages 149-153
doi:10.1155/S0161171201006251
On some properties of Banach operators
Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
Received 16 November 2000
Copyright © 2001 A. B. Thaheem and AbdulRahim Khan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.