SELF-ADJOINT INTERPOLATION FOR OPERATORS IN TRIDIAGONAL ALGEBRAS

Title & Authors
SELF-ADJOINT INTERPOLATION FOR OPERATORS IN TRIDIAGONAL ALGEBRAS
Kang, Joo-Ho; Jo, Young-Soo;

Abstract
Given operators X and Y acting on a Hilbert space H, an interpolating operator is a bounded operator A such that AX = Y. An interpolating operator for n-operators satisfies the equation $\small{AX_{}i}$ = $\small{Y_{i}}$ for i/ = 1,2,…, n. In this article, we obtained the following : Let X ＝ ($\small{x_{i\sigma(i)}}$ and Y ＝ ($\small{y_{ij}}$ be operators in B(H) such that $\small{X_{i\sigma(i)}\neq\;0}$ for all i. Then the following statements are equivalent. (1) There exists an operator A in Alg L such that AX = Y, every E in L reduces A and A is a self-adjoint operator. (2) sup $\small{{\frac{\parallel{\sum^n}_{i=1}E_iYf_i\parallel}{\parallel{\sum^n}_{i=1}E_iXf_i\parallel}n\;\epsilon\;N,E_i\;\epsilon\;L and f_i\;\epsilon\;H}}$ < $\small{\infty}$ and $\small{x_{i,\sigma(i)}y_{i,\sigma(i)}}$ is real for all i = 1,2, ....
Keywords
Language
English
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