Open Access

Extensions of the results on powers of https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq1_HTML.gif -hyponormal and https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq2_HTML.gif -hyponormal operators

Journal of Inequalities and Applications20062006:36919

DOI: 10.1155/JIA/2006/36919

Received: 22 November 2004

Accepted: 10 May 2005

Published: 9 March 2006

Abstract

Firstly, we will show the following extension of the results on powers of https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq3_HTML.gif -hyponormal and https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq4_HTML.gif -hyponormal operators: let https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq6_HTML.gif be positive integers, if https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq7_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq8_HTML.gif -hyponormal for https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq9_HTML.gif , then: (i) in case https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq10_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq11_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq12_HTML.gif hold, (ii) in case https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq13_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq14_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq15_HTML.gif hold. Secondly, we will show an estimation on powers of https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq16_HTML.gif -hyponormal operators for https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq17_HTML.gif which implies the best possibility of our results. Lastly, we will show a parallel estimation on powers of https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq18_HTML.gif -hyponormal operators as follows: let https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq19_HTML.gif , then the following hold for each positive integer https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq20_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq21_HTML.gif : (i) there exists a log-hyponormal operator https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq22_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq23_HTML.gif , (ii) there exists a https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq24_HTML.gif -hyponormal operator https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq25_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F36919/MediaObjects/13660_2004_Article_1588_IEq26_HTML.gif .

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Authors’ Affiliations

(1)
College of Mathematics and Information Science, Henan Normal University

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Copyright

© C. Yang and J. Yuan 2006

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