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Journal of Inequalities and Applications

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Open Access 01-12-2006 | Research Article

A characterization of chaotic order

Authors: Changsen Yang, Fugen Gao

Published in: Journal of Inequalities and Applications | Issue 1/2006

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Abstract

The chaotic order

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq1_HTML.gif

among positive invertible operators

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq2_HTML.gif

on a Hilbert space is introduced by

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq3_HTML.gif

. Using Uchiyama's method and Furuta's Kantorovich-type inequality, we will point out that

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq4_HTML.gif

if and only if

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq5_HTML.gif

holds for any

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq6_HTML.gif

, where

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq7_HTML.gif

is any fixed positive number. On the other hand, for any fixed

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq8_HTML.gif

, we also show that there exist positive invertible operators

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq9_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq10_HTML.gif

such that

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq11_HTML.gif

holds for any

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq12_HTML.gif

, but

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq13_HTML.gif

is not valid.

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Title: A characterization of chaotic order
Authors: Changsen Yang
Fugen Gao
Publication date: 01-12-2006
Publisher: Springer International Publishing
Published in: Journal of Inequalities and Applications / Issue 1/2006
Electronic ISSN: 1029-242X
DOI: https://doi.org/10.1155/JIA/2006/79123

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