SELF-ADJOINT INTERPOLATION ON AX = Y IN $\small{\mathcal{B}(\mathcal{H})}$

• Journal title : Honam Mathematical Journal
• Volume 30, Issue 4,  2008, pp.685-691
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2008.30.4.685
Title & Authors
SELF-ADJOINT INTERPOLATION ON AX = Y IN $\small{\mathcal{B}(\mathcal{H})}$
Kwak, Sung-Kon; Kim, Ki-Sook;

Abstract
Given operators $\small{X_i}$ and $\small{Y_i}$ (i = 1, 2, $\small{{\cdots}}$, n) acting on a Hilbert space $\small{\mathcal{H}}$, an interpolating operator is a bounded operator A acting on $\small{\mathcal{H}}$ such that $\small{AX_i}$ = $\small{Y_i}$ for i= 1, 2, $\small{{\cdots}}$, n. In this article, if the range of $\small{X_k}$ is dense in H for a certain k in {1, 2, $\small{{\cdots}}$, n), then the following are equivalent: (1) There exists a self-adjoint operator A in $\small{\mathcal{B}(\mathcal{H})}$ stich that $\small{AX_i}$ = $\small{Y_i}$ for I = 1, 2, $\small{{\cdots}}$, n. (2) $\small{sup\{{\frac{{\parallel}{\sum}^n_{i=1}Y_if_i{\parallel}}{{\parallel}{\sum}^n_{i=1}X_if_i{\parallel}}:f_i{\in}H}\}}$ < $\small{{\infty}}$ and < $\small{X_kf,Y_kg}$ >=< $\small{Y_kf,X_kg}$> for all f, g in $\small{\mathcal{H}}$.
Keywords
Language
English
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