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

• Journal title : Honam Mathematical Journal
• Volume 31, Issue 3,  2009, pp.421-428
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2009.31.3.421
Title & Authors
UNITARY INTERPOLATION ON AX = Y IN ALG$\small{\mathcal{L}}$
Kang, Joo-Ho;

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 paper, we showed the following : Let $\small{\mathcal{L}}$ be a subspace lattice acting on a Hilbert space $\small{\mathcal{H}}$ and let $\small{X_i}$ and $\small{Y_i}$ be operators in B($\small{\mathcal{H}}$) for i = 1, 2, $\small{{\cdots}}$. Let $\small{P_i}$ be the projection onto $\small{\overline{rangeX_i}}$ for all i = 1, 2, $\small{{\cdots}}$. If $\small{P_kE}$ = $\small{EP_k}$ for some k in $\small{\mathbb{N}}$ and all E in $\small{\mathcal{L}}$, then the following are equivalent: (1) $\small{sup\;\{{\frac{{\parallel}E^{\perp}({\sum}^n_{i=1}Y_if_i){\parallel}}{{\parallel}E^{\perp}({\sum}^n_{i=1}Y_if_i){\parallel}}:f{\in}H,n{\in}{\mathbb{N}},E{\in}\mathcal{L}}\}}$ < $\small{{\infty}}$ range $\small{\overline{rangeY_k}\;=\;\overline{rangeX_k}\;=\;\mathcal{H}}$, and < $\small{X_kf,\;X_kg}$ >=< $\small{Y_kf,\;Y_kg}$ > for some k in $\small{\mathbb{N}}$ and for all f and g in $\small{\mathcal{H}}$. (2) There exists an operator A in Alg$\small{\mathcal{L}}$ such that $\small{AX_i}$ = $\small{Y_i}$ for i = 1, 2, $\small{{\cdots}}$ and AA$\small{^*}$ = I = A$\small{^*}$A.
Keywords
Interpolaion Problem;Unitary Interpolation Problem;Subspace Lattice;Alg $\small{\mathcal{L}}$;
Language
English
Cited by
References
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