INTERPOLATION PROBLEMS FOR OPERATORS WITH CORANK IN ALG L

• Journal title : Honam Mathematical Journal
• Volume 34, Issue 3,  2012, pp.409-422
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2012.34.3.409
Title & Authors
INTERPOLATION PROBLEMS FOR OPERATORS WITH CORANK IN ALG L
Kang, Joo-Ho;

Abstract
Let $\small{\mathcal{L}}$ be a subspace lattice on a Hilbert space $\small{\mathcal{H}}$. And let X and Y be operators acting on a Hilbert space $\small{\mathcal{H}}$. Let $\small{sp(x)=\{{\alpha}x\;:\;{\alpha}{\in}\mathcal{C}\}}$ $\small{x{\in}\mathcal{H}}$. Assume that $\small{\mathcal{H}=\overline{range\;X}{\oplus}sp(h)}$ for some $\small{h{\in}\mathcal{H}}$ and < $\small{h}$, $\small{E^{\bot}Xf}$ >= 0 for each $\small{f{\in}\mathcal{H}}$ and $\small{E{\in}\mathcal{L}}$. Then there exists an operator A in Alg$\small{\mathcal{L}}$ such that AX = Y if and only if $\small{sup\{\frac{{\parallel}E^{\bot}Yf{\parallel}}{{\parallel}E^{\bot}Yf{\parallel}}\;:\;f{\in}H,\;E{\in}\mathcal{L}\}}$ = K < $\small{{\infty}}$. Moreover, if the necessary condition holds, then we may choose an operator A such that AX = Y and $\small{{\parallel}||A{\parallel}=K}$.
Keywords
Interpolation Problem;Subspace Lattice;Alg$\small{\mathcal{L}}$;CSL-Alg$\small{\mathcal{L}}$;
Language
English
Cited by
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