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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A perturbed elementary operator and range-kernel orthogonality


Author: B. P. Duggal
Journal: Proc. Amer. Math. Soc. 134 (2006), 1727-1734
MSC (2000): Primary 47B47, 47B10, 47A10, 47B40
Published electronically: December 19, 2005
MathSciNet review: 2204285
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Abstract: Let $ B(\mathcal{H})$ denote the algebra of operators on a Hilbert $ \mathcal{H}$. If $ A_j$ and $ B_j\in B(\mathcal{H})$ are commuting normal operators, and $ C_j$ and $ D_j\in B(\mathcal{H})$ are commuting quasi-nilpotents such that $ A_jC_j-C_jA_j=B_jD_j-D_jB_j=0$, then define $ M_j, N_j\in B(\mathcal{H})$ and $ {\mathcal E}, E\in B(B(\mathcal{H}))$ by $ M_j=A_j+C_j$, $ N_j=B_j+D_j$, $ {\mathcal E}(X)=A_1XA_2+B_1XB_2$ and $ E(X)=M_1XM_2+N_1XN_2$. It is proved that $ E^{-1}(0)\subseteq H_0({\mathcal E})={\mathcal E}^{-1}(0)$ and $ X\in E^{-1}(0)\Longrightarrow \vert\vert X\vert\vert\leq k \textrm{dist}(X, {\mathcal E}(B(\mathcal{H})))$, where $ k\geq 1$ is some scalar and $ H_0({\mathcal E})$ is the quasi-nilpotent part of the operator $ {\mathcal E}$.


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Additional Information

B. P. Duggal
Affiliation: Department of Mathematics, College of Science UAEU, P.O. Box 17551, Al Ain, United Arab Emirates
Address at time of publication: 8 Redwood Grove, Northfield Avenue, London W5 4SZ, United Kingdom
Email: bpduggal@uaeu.ac.ae, bpduggal@yahoo.co.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08337-1
PII: S 0002-9939(05)08337-1
Keywords: Hilbert space, elementary operator, normal operator, quasi-nilpotent operator, generalized scalar operator, orthogonality
Received by editor(s): June 29, 2004
Received by editor(s) in revised form: January 14, 2005
Published electronically: December 19, 2005
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.