Trace-class interpolation for vectors in tridiagonal algebras
Bull. Korean Math. Soc. 2002 Vol. 39, No. 1, 63-69
Published online March 1, 2002
Young Soo Jo and Joo Ho Kang
Keimyung University, Taegu University
Abstract : Given vectors $x$ and $y$ in a Hilbert space, an interpolating operator is a bounded operator $T$ such that $Tx=y$. An interpolating operator for $n$ vectors satisfies the equation $Tx_i =y_i$, for $i=1,2,\cdots,n$. In this article, we obtained the following : Let $x=(x_i)$ and $y=(y_i)$ be two vectors in $\Cal H$ such that $x_i \neq 0$ for all $i=1,2,\cdots$. Then the following statements are equivalent. \itemitem{\rm (1)} There exists an operator $A$ in Alg$\Cal L$ such that $Ax =y$, $A$ is a trace-class operator and every $E$ in $\Cal L$ reduces $A$. \itemitem{\rm (2)} $\displaystyle \sup \left\{ {{\| \sum_{k=1}^l \alpha_k E_k y\|} \over {\| \sum_{k=1}^l \alpha_k E_k x\|}} : l \in N, \alpha_k \in {\Bbb C} \text{~and~} E_k \in {\Cal L} \right\} < \infty$ \linebreak and $\displaystyle\sum_{n=1}^\infty |y_n| |x_n|^{-1} < \infty$.
Keywords : trace-class, tridiagonal algebra, commutative subspace lattice, Alg$\Cal L$
MSC numbers : 47L35
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