SELF-ADJOINT INTERPOLATION ON AX = Y IN ALGL

• Journal title : Honam Mathematical Journal
• Volume 29, Issue 1,  2007, pp.55-60
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2007.29.1.055
Title & Authors
SELF-ADJOINT INTERPOLATION ON AX = Y IN ALGL
Jo, Young-Soo; Kang, Joo-Ho;

Abstract
Given operators X and Y acting on a Hilbert space $\small{\cal{H}}$, an interpolating operator is a bounded operator A such that AX = Y. In this article, we showed the following : Let $\small{\cal{L}}$ be a subspace lattice acting on a Hilbert space $\small{\cal{H}}$ and let X and Y be operators in $\small{\cal{B}(\cal{H})}$. Let P be the projection onto $\small{\bar{rangeX}}$. If FE = EF for every $\small{E\in\cal{L}}$, then the following are equivalent: (1) $\small{sup\{{{\parallel}E^{\perp}Yf\parallel\atop \parallel{E}^{\perp}Xf\parallel}\;:\;f{\in}\cal{H},\;E\in\cal{L}\}\}$ < $\small{\infty}$, $\small{\bar{range\;Y}\subset\bar{range\;X}}$, and < Xf, Yg >=< Yf,Xg > for any f and g in $\small{\cal{H}}$. (2) There exists a self-adjoint operator A in Alg$\small{\cal{L}}$ such that AX = Y.
Keywords
Interpolation Problem;Self-Adjoint Interpolation Problem;Subspace Lattice;Alg $\small{\cal{L}}$.;
Language
English
Cited by
References
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