Self-adjoint interpolation for operators in tridiagonal algebras

Abstract

Given operators $X$ and $Y$ acting on a Hilbert space $\Cal H$, an interpolating operator is a bounded operator $A$ such that $AX=Y$. An interpolating operator for $n$-operators satisfies the equation $AX_i=Y_i$ for $i=1,2,\cdots,n$. In this article, we obtained the following : Let $X = (x_{ij})$ and $Y =(y_{ij})$ be operators in ${\Cal B}({\Cal H})$ such that $x_{i \sigma(i)} \neq 0$ for all $i$. Then the following statements are equivalent. (1) There exists an operator $A$ in Alg$\Cal L$ such that $AX =Y$, every $E$ in $\Cal L$ reduces $A$ and $A$ is a self-adjoint operator. (2) $\displaystyle \sup \left\{ {{\| \sum_{i=1}^n E_i Y f_i\|} \over{\| \sum_{i=1}^n E_i X f_i\|}} : n \in {\Bbb N}, E_i \in {\Cal L} \text{~and~} f_i \in {\Cal H}\right\} < \infty$ \linebreak and ${\overline {x_{i,\sigma(i)}}} y_{i,\sigma(i)}$ is real for all $i=1,2,\cdots$.