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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| Format: | Article |
| Language: | English |
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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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| _version_ | 1832556835012345856 |
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| author | A. B. Thaheem AbdulRahim Khan |
| author_facet | A. B. Thaheem AbdulRahim Khan |
| author_sort | A. B. Thaheem |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-94034e637a5d434588c9a30f5f69dd23 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2001-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-94034e637a5d434588c9a30f5f69dd232025-02-03T05:44:16ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252001-01-0127314915310.1155/S0161171201006251On some properties of Banach operatorsA. B. Thaheem0AbdulRahim Khan1Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi ArabiaDepartment of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi ArabiaA 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.http://dx.doi.org/10.1155/S0161171201006251 |
| spellingShingle | A. B. Thaheem AbdulRahim Khan On some properties of Banach operators International Journal of Mathematics and Mathematical Sciences |
| title | On some properties of Banach operators |
| title_full | On some properties of Banach operators |
| title_fullStr | On some properties of Banach operators |
| title_full_unstemmed | On some properties of Banach operators |
| title_short | On some properties of Banach operators |
| title_sort | on some properties of banach operators |
| url | http://dx.doi.org/10.1155/S0161171201006251 |
| work_keys_str_mv | AT abthaheem onsomepropertiesofbanachoperators AT abdulrahimkhan onsomepropertiesofbanachoperators |