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

 

Interpolation in self-adjoint settings


Authors: Y. S. Jo, J. H. Kang, R. L. Moore and T. T. Trent
Journal: Proc. Amer. Math. Soc. 130 (2002), 3269-3281
MSC (2000): Primary 46L10, 47L35
Published electronically: June 11, 2002
MathSciNet review: 1913006
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Abstract: We study the operator equation $AX=Y$, where the operators $X$ and $Y$are given and the operator $A$ is required to lie in some von Neumann algebra. We derive a necessary and sufficient condition for the existence of a solution $A$. The condition is that there must exist a constant $K$ so that, for all finite collections of operators $\{D_{1},D_{2}, \dots , D_{n}\}$ in the commutant, and all collections of vectors $\{f_{1}, f_{2}, \dots , f_{n}\}$, we have $\Vert\sum _{j=1}^{n} D_{j} Y f_{j} \Vert\leq K\, \Vert\sum _{j=1}^{n} D_{j} X f_{j} \Vert\;. $We also study the equality $\Vert DYf\Vert= K\Vert DXf\Vert$, in connection with solving the equation $AX=Y$ where the operator $A$ is required to lie in some CSL algebra.


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Additional Information

Y. S. Jo
Affiliation: Department of Mathematics, Keimyung University, Taegu, Korea

J. H. Kang
Affiliation: Department of Mathematics, Taegu University, Taegu 712-714, Korea

R. L. Moore
Affiliation: Department of Mathematics, Box 870350, University of Alabama, Tuscaloosa, Alabama 35487-0350
Email: rmoore@gp.as.ua.edu

T. T. Trent
Affiliation: Department of Mathematics, Box 870350, University of Alabama, Tuscaloosa, Alabama 35487-0350
Email: ttrent@gp.as.ua.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06610-8
PII: S 0002-9939(02)06610-8
Received by editor(s): September 1, 2000
Received by editor(s) in revised form: June 7, 2001
Published electronically: June 11, 2002
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society