Some Estimates of Certain Subnormal and Hyponormal Derivations

We prove that if 𝐴 and 𝐵∗ are subnormal operators and 𝑋 is a bounded linear operator such that 𝐴𝑋−𝑋𝐵 is a Hilbert-Schmidt operator, then 𝑓(𝐴)𝑋−𝑋𝑓(𝐵) is also a Hilbert-Schmidt operator and ‖𝑓(𝐴)𝑋−𝑋𝑓(𝐵)‖2≤𝐿‖𝐴𝑋−𝑋𝐵‖2 for 𝑓 belongs to a certain class of functions. Furthermore, we investigate the simil...

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Main Author: Vasile Lauric
Format: Article
Language:English
Published: Hindawi Limited 2008-01-01
Series:International Journal of Mathematics and Mathematical Sciences
Online Access:http://dx.doi.org/10.1155/2008/362409
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spelling doaj-6294783083224e9b87feb786d45bda132020-11-24T22:57:03ZengHindawi LimitedInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252008-01-01200810.1155/2008/362409362409Some Estimates of Certain Subnormal and Hyponormal DerivationsVasile Lauric0Department of Mathematics, Florida A&M University, Tallahassee, FL 32307, USAWe prove that if 𝐴 and 𝐵∗ are subnormal operators and 𝑋 is a bounded linear operator such that 𝐴𝑋−𝑋𝐵 is a Hilbert-Schmidt operator, then 𝑓(𝐴)𝑋−𝑋𝑓(𝐵) is also a Hilbert-Schmidt operator and ‖𝑓(𝐴)𝑋−𝑋𝑓(𝐵)‖2≤𝐿‖𝐴𝑋−𝑋𝐵‖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∈ℒ(ℋ) is such that 𝑆𝑋−𝑋𝑇 belongs to a norm ideal (𝐽,‖⋅‖𝐽), and we prove that 𝑓(𝑆)𝑋−𝑋𝑓(𝑇)∈𝐽 and ‖𝑓(𝑆)𝑋−𝑋𝑓(𝑇)‖𝐽≤𝐶‖𝑆𝑋−𝑋𝑇‖𝐽 for 𝑓 being in a certain class of functions.http://dx.doi.org/10.1155/2008/362409
collection DOAJ
language English
format Article
sources DOAJ
author Vasile Lauric
spellingShingle Vasile Lauric
Some Estimates of Certain Subnormal and Hyponormal Derivations
International Journal of Mathematics and Mathematical Sciences
author_facet Vasile Lauric
author_sort Vasile Lauric
title Some Estimates of Certain Subnormal and Hyponormal Derivations
title_short Some Estimates of Certain Subnormal and Hyponormal Derivations
title_full Some Estimates of Certain Subnormal and Hyponormal Derivations
title_fullStr Some Estimates of Certain Subnormal and Hyponormal Derivations
title_full_unstemmed Some Estimates of Certain Subnormal and Hyponormal Derivations
title_sort some estimates of certain subnormal and hyponormal derivations
publisher Hindawi Limited
series International Journal of Mathematics and Mathematical Sciences
issn 0161-1712
1687-0425
publishDate 2008-01-01
description We prove that if 𝐴 and 𝐵∗ are subnormal operators and 𝑋 is a bounded linear operator such that 𝐴𝑋−𝑋𝐵 is a Hilbert-Schmidt operator, then 𝑓(𝐴)𝑋−𝑋𝑓(𝐵) is also a Hilbert-Schmidt operator and ‖𝑓(𝐴)𝑋−𝑋𝑓(𝐵)‖2≤𝐿‖𝐴𝑋−𝑋𝐵‖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∈ℒ(ℋ) is such that 𝑆𝑋−𝑋𝑇 belongs to a norm ideal (𝐽,‖⋅‖𝐽), and we prove that 𝑓(𝑆)𝑋−𝑋𝑓(𝑇)∈𝐽 and ‖𝑓(𝑆)𝑋−𝑋𝑓(𝑇)‖𝐽≤𝐶‖𝑆𝑋−𝑋𝑇‖𝐽 for 𝑓 being in a certain class of functions.
url http://dx.doi.org/10.1155/2008/362409
work_keys_str_mv AT vasilelauric someestimatesofcertainsubnormalandhyponormalderivations
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