NORMAL INTERPOLATION ON AX = Y IN ALG$\small{\mathcal{L}}$

• Journal title : Honam Mathematical Journal
• Volume 30, Issue 2,  2008, pp.329-334
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2008.30.2.329
Title & Authors
NORMAL INTERPOLATION ON AX = Y IN ALG$\small{\mathcal{L}}$
Jo, Young-Soo;

Abstract
Given operators X and Y acting on a Hilbert space $\small{\mathcal{H}}$, an interpolating operator is a bounded operator A such that AX = Y. In this article, the following is proved: Let $\small{\mathcal{L}}$ be a subspace lattice on $\small{\mathcal{H}}$ and let X and Y be operators acting on a Hilbert space H. Let P be the projection onto the $\small{\overline{rangeX}}$. If PE = EP for each E $\small{{\in}}$ $\small{\mathcal{L}}$, then the following are equivalent: (1) sup $\small{{{\frac{{\parallel}E^{\perp}Yf{\parallel}}{{\parallel}E^{\perp}Xf{\parallel}}}:f{\in}\mathcal{H},\;E{\in}\mathcal{L}}}$ < $\small{{\infty},\;\overline{rangeY}\;{\subset}\;\overline{rangeX}}$, and there is a bounded operator T acting on $\small{\mathcal{H}}$ such that < Xf, Tg >=< Yf, Xg >, < Tf, Tg >=< Yf, Yg > for all f and gin $\small{\mathcal{H}}$ and $\small{T^*h}$ = 0 for h $\small{{\in}\;{\overline{rangeX}}^{\perp}}$. (2) There is a normal operator A in AlgL such that AX = Y and Ag = 0 for all g in range $\small{{\overline{rangeX}}^{\perp}}$.
Keywords
Interpolation Problem;Normal Interpolation Problem;Subspace Lattice;Alg $\small{\mathcal{L}}$;
Language
English
Cited by
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