A perturbed elementary operator and range-kernel orthogonality
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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 $X\in E^{-1}(0)\Longrightarrow ||X||\leq k \textrm {dist}(X, {\mathcal E}(B(\mathcal {H})))$, where $k\geq 1$ is some scalar and $H_0({\mathcal E})$ is the quasi-nilpotent part of the operator ${\mathcal E}$.References
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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
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 1727-1734
- MSC (2000): Primary 47B47, 47B10, 47A10, 47B40
- DOI: https://doi.org/10.1090/S0002-9939-05-08337-1
- MathSciNet review: 2204285