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 spac...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2001-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171201006251 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |