Journal of Inequalities and Applications
Volume 2011 (2011), Article ID 564836, 14 pages
doi:10.1155/2011/564836
Research Article

Some Vector Inequalities for Continuous Functions of Self-Adjoint Operators in Hilbert Spaces

S. S. Dragomir1,2

1Mathematics, School of Engineering & Science, Victoria University, P.O. Box 14428, Melbourne City, VIC 8001, Australia
2School of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag-3, Johannesburg 2050, South Africa

Received 24 November 2010; Accepted 21 February 2011

Academic Editor: Paolo E. Ricci

Copyright © 2011 S. S. Dragomir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

On utilizing the spectral representation of self-adjoint operators in Hilbert spaces, some inequalities for the composite operator [ 𝑓 ( 𝑀 ) 1 𝐻 − 𝑓 ( 𝐴 ) ] [ 𝑓 ( 𝐴 ) − 𝑓 ( 𝑚 ) 1 𝐻 ] , where S p ( 𝐴 ) ⊆ [ 𝑚 , 𝑀 ] and for various classes of continuous functions 𝑓 ∶ [ 𝑚 , 𝑀 ] → ℂ are given. Applications for the power function and the logarithmic function are also provided.