POSITIVE INTERPOLATION ON Ax = y AND AX = Y IN ALG$\small{\mathcal{L}}$

• Journal title : Honam Mathematical Journal
• Volume 31, Issue 2,  2009, pp.259-265
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2009.31.2.259
Title & Authors
POSITIVE INTERPOLATION ON Ax = y AND AX = Y IN ALG$\small{\mathcal{L}}$
Kang, Joo-Ho;

Abstract
Let $\small{\mathcal{L}}$ be a subspace lattice on a Hilbert space $\small{\mathcal{H}}$. Let x and y be vectors in $\small{\mathcal{H}}$ and let $\small{P_x}$ be the projection onto sp(x). If $\small{P_xE}$ = $\small{EP_x}$ for each E $\small{{\in}\;\mathcal{L}}$, then the following are equivalent. (1) There exists an operator A in Alg$\small{\mathcal{L}}$ such that Ax = y, Af = 0 for all f in $\small{sp(x)^{\perp}}$ and A $\small{{\geq}}$ 0. (2) sup $\small{{\frac{{\parallel}E^{\perp}y{\parallel}}{{\parallel}E^{\perp}x{\parallel}}:E{\in}\mathcal{L}}}$ < $\small{{\infty}}$ < x, y > $\small{{\geq}}$ 0. Let X and Y be operators in $\small{\mathcal{B}(\mathcal{H})}$. Let P be the projection onto $\small{\overline{rangeX}}$. If PE = EP for each E $\small{{\in}\;\mathcal{L}}$, then the following are equivalent: (1) sup $\small{{\frac{{\parallel}E^{\perp}Yf{\parallel}}{{\parallel}E^{\perp}Xf{\parallel}}:f{\in}\mathcal{H},E{\in}\mathcal{L}}}$ < $\small{{\infty}}$ and < Xf, Yf > $\small{{\geq}}$ 0 for all f in H. (2) There exists a positive operator A in Alg$\small{\mathcal{L}}$ such that AX = Y.
Keywords
Interpolation Problem;Subspace Lattice;Positive Interpolation Problem;Alg$\small{\mathcal{L}}$;
Language
English
Cited by
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