On some properties of Banach operators

A. B. Thaheem and AbdulRahim Khan

International Journal of Mathematics and Mathematical Sciences, 2001, vol. 27, 1-5

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 ‖ α 2 ( 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.

Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:198437

DOI: 10.1155/S0161171201006251

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