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International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 362409, 6 pages
http://dx.doi.org/10.1155/2008/362409
Research Article

Some Estimates of Certain Subnormal and Hyponormal Derivations

Department of Mathematics, Florida A&M University, Tallahassee, FL 32307, USA

Received 23 October 2007; Accepted 9 January 2008

Academic Editor: Nils Ackermann

Copyright © 2008 Vasile Lauric. 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

We prove that if 𝐴 and 𝐡 βˆ— are subnormal operators and 𝑋 is a bounded linear operator such that 𝐴 𝑋 βˆ’ 𝑋 𝐡 is a Hilbert-Schmidt operator, then 𝑓 ( 𝐴 ) 𝑋 βˆ’ 𝑋 𝑓 ( 𝐡 ) is also a Hilbert-Schmidt operator and β€– 𝑓 ( 𝐴 ) 𝑋 βˆ’ 𝑋 𝑓 ( 𝐡 ) β€– 2 ≀ 𝐿 β€– 𝐴 𝑋 βˆ’ 𝑋 𝐡 β€– 2 for 𝑓 belongs to a certain class of functions. Furthermore, we investigate the similar problem in the case that 𝑆 , 𝑇 are hyponormal operators and 𝑋 ∈ β„’ ( β„‹ ) is such that 𝑆 𝑋 βˆ’ 𝑋 𝑇 belongs to a norm ideal ( 𝐽 , β€– β‹… β€– 𝐽 ) , and we prove that 𝑓 ( 𝑆 ) 𝑋 βˆ’ 𝑋 𝑓 ( 𝑇 ) ∈ 𝐽 and β€– 𝑓 ( 𝑆 ) 𝑋 βˆ’ 𝑋 𝑓 ( 𝑇 ) β€– 𝐽 ≀ 𝐢 β€– 𝑆 𝑋 βˆ’ 𝑋 𝑇 β€– 𝐽 for 𝑓 being in a certain class of functions.