Interpolation of compact non-linear operators
<p/> <p>Let <inline-formula><graphic file="1029-242X-2000-862170-i1.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-2000-862170-i2.gif"/></inline-formula> be two Banach couples and let <inline-formula><...
| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2000-01-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://www.journalofinequalitiesandapplications.com/content/5/862170 |
| _version_ | 1828783744530513920 |
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| author | Bento AJG |
| author_facet | Bento AJG |
| author_sort | Bento AJG |
| collection | DOAJ |
| description | <p/> <p>Let <inline-formula><graphic file="1029-242X-2000-862170-i1.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-2000-862170-i2.gif"/></inline-formula> be two Banach couples and let <inline-formula><graphic file="1029-242X-2000-862170-i3.gif"/></inline-formula> be a continuous map such that <inline-formula><graphic file="1029-242X-2000-862170-i4.gif"/></inline-formula> is a Lipschitz compact operator and <inline-formula><graphic file="1029-242X-2000-862170-i5.gif"/></inline-formula> is a Lipschitz operator. We prove that if <inline-formula><graphic file="1029-242X-2000-862170-i6.gif"/></inline-formula> is also compact or <inline-formula><graphic file="1029-242X-2000-862170-i7.gif"/></inline-formula> is continuously embedded in <inline-formula><graphic file="1029-242X-2000-862170-i8.gif"/></inline-formula> or <inline-formula><graphic file="1029-242X-2000-862170-i9.gif"/></inline-formula> is continuously embedded in <inline-formula><graphic file="1029-242X-2000-862170-i10.gif"/></inline-formula>, then <inline-formula><graphic file="1029-242X-2000-862170-i11.gif"/></inline-formula> is also a compact operator when <inline-formula><graphic file="1029-242X-2000-862170-i12.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-2000-862170-i13.gif"/></inline-formula>. We also investigate the behaviour of the measure of non-compactness under real interpolation and obtain best possible compactness results of Lions–Peetre type for non-linear operators. A two-sided compactness result for linear operators is also obtained for an arbitrary interpolation method when an approximation hypothesis on the Banach couple <inline-formula><graphic file="1029-242X-2000-862170-i14.gif"/></inline-formula> is imposed.</p> |
| first_indexed | 2024-12-11T23:18:38Z |
| format | Article |
| id | doaj.art-7bfc5bf4a62b4985ac9a6871f26ef48d |
| institution | Directory Open Access Journal |
| issn | 1025-5834 1029-242X |
| language | English |
| last_indexed | 2024-12-11T23:18:38Z |
| publishDate | 2000-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-7bfc5bf4a62b4985ac9a6871f26ef48d2022-12-22T00:46:26ZengSpringerOpenJournal of Inequalities and Applications1025-58341029-242X2000-01-0120003862170Interpolation of compact non-linear operatorsBento AJG<p/> <p>Let <inline-formula><graphic file="1029-242X-2000-862170-i1.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-2000-862170-i2.gif"/></inline-formula> be two Banach couples and let <inline-formula><graphic file="1029-242X-2000-862170-i3.gif"/></inline-formula> be a continuous map such that <inline-formula><graphic file="1029-242X-2000-862170-i4.gif"/></inline-formula> is a Lipschitz compact operator and <inline-formula><graphic file="1029-242X-2000-862170-i5.gif"/></inline-formula> is a Lipschitz operator. We prove that if <inline-formula><graphic file="1029-242X-2000-862170-i6.gif"/></inline-formula> is also compact or <inline-formula><graphic file="1029-242X-2000-862170-i7.gif"/></inline-formula> is continuously embedded in <inline-formula><graphic file="1029-242X-2000-862170-i8.gif"/></inline-formula> or <inline-formula><graphic file="1029-242X-2000-862170-i9.gif"/></inline-formula> is continuously embedded in <inline-formula><graphic file="1029-242X-2000-862170-i10.gif"/></inline-formula>, then <inline-formula><graphic file="1029-242X-2000-862170-i11.gif"/></inline-formula> is also a compact operator when <inline-formula><graphic file="1029-242X-2000-862170-i12.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-2000-862170-i13.gif"/></inline-formula>. We also investigate the behaviour of the measure of non-compactness under real interpolation and obtain best possible compactness results of Lions–Peetre type for non-linear operators. A two-sided compactness result for linear operators is also obtained for an arbitrary interpolation method when an approximation hypothesis on the Banach couple <inline-formula><graphic file="1029-242X-2000-862170-i14.gif"/></inline-formula> is imposed.</p>http://www.journalofinequalitiesandapplications.com/content/5/862170InterpolationCompact non-linear operatorsMeasure of non-compactness |
| spellingShingle | Bento AJG Interpolation of compact non-linear operators Journal of Inequalities and Applications Interpolation Compact non-linear operators Measure of non-compactness |
| title | Interpolation of compact non-linear operators |
| title_full | Interpolation of compact non-linear operators |
| title_fullStr | Interpolation of compact non-linear operators |
| title_full_unstemmed | Interpolation of compact non-linear operators |
| title_short | Interpolation of compact non-linear operators |
| title_sort | interpolation of compact non linear operators |
| topic | Interpolation Compact non-linear operators Measure of non-compactness |
| url | http://www.journalofinequalitiesandapplications.com/content/5/862170 |
| work_keys_str_mv | AT bentoajg interpolationofcompactnonlinearoperators |