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: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2000-01-01
|
| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://www.journalofinequalitiesandapplications.com/content/5/862170 |
| Summary: | <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> |
|---|---|
| ISSN: | 1025-5834 1029-242X |