A characterization of chaotic order
<p/> <p>The chaotic order <inline-formula><graphic file="1029-242X-2006-79123-i1.gif"/></inline-formula> among positive invertible operators <inline-formula><graphic file="1029-242X-2006-79123-i2.gif"/></inline-formula> on a Hilbert...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2006-01-01
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| Series: | Journal of Inequalities and Applications |
| Online Access: | http://www.journalofinequalitiesandapplications.com/content/2006/79123 |
| Summary: | <p/> <p>The chaotic order <inline-formula><graphic file="1029-242X-2006-79123-i1.gif"/></inline-formula> among positive invertible operators <inline-formula><graphic file="1029-242X-2006-79123-i2.gif"/></inline-formula> on a Hilbert space is introduced by <inline-formula><graphic file="1029-242X-2006-79123-i3.gif"/></inline-formula>. Using Uchiyama's method and Furuta's Kantorovich-type inequality, we will point out that <inline-formula><graphic file="1029-242X-2006-79123-i4.gif"/></inline-formula> if and only if <inline-formula><graphic file="1029-242X-2006-79123-i5.gif"/></inline-formula> holds for any <inline-formula><graphic file="1029-242X-2006-79123-i6.gif"/></inline-formula>, where <inline-formula><graphic file="1029-242X-2006-79123-i7.gif"/></inline-formula> is any fixed positive number. On the other hand, for any fixed <inline-formula><graphic file="1029-242X-2006-79123-i8.gif"/></inline-formula>, we also show that there exist positive invertible operators <inline-formula><graphic file="1029-242X-2006-79123-i9.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2006-79123-i10.gif"/></inline-formula> such that <inline-formula><graphic file="1029-242X-2006-79123-i11.gif"/></inline-formula> holds for any <inline-formula><graphic file="1029-242X-2006-79123-i12.gif"/></inline-formula>, but <inline-formula><graphic file="1029-242X-2006-79123-i13.gif"/></inline-formula> is not valid.</p> |
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| ISSN: | 1025-5834 1029-242X |