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ISSN : 3059-0604
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HILBERT-SCHMIDT INTERPOLATION ON AX=Y IN A TRIDIAGONAL ALGEBRA ALG<TEX>${\pounds}$</TEX>
HILBERT-SCHMIDT INTERPOLATION ON AX = Y IN A TRIDIAGONAL ALGEBRA ALGL
Abstract
Given operators X and Y acting on a separable complex Hilbert space H, an interpolating operator is a bounded operator A such that AX=Y. In this article, we investigate Hilbert-Schmidt interpolation problems for operators in a tridiagonal algebra and we get the following: Let <TEX>${\pounds}$</TEX> be a subspace lattice acting on a separable complex Hilbert space H and let X=<TEX>$(x_{ij})$</TEX> and Y=<TEX>$(y_{ij})$</TEX> be operators acting on H. Then the following are equivalent: (1) There exists a Hilbert-Schmidt operator <TEX>$A=(a_{ij})$</TEX> in Alg<TEX>${\pounds}$</TEX> such that AX=Y. (2) There is a bounded sequence <TEX>$\{{\alpha}_n\}$</TEX> in <TEX>$\mathbb{C}$</TEX> such that <TEX>${\sum}_{n=1}^{\infty}|{\alpha}_n|^2</TEX><TEX><</TEX><TEX>{\infty}$</TEX> and <TEX>
- keywords
- Hilbert-Schmidt interpolation, CSL-algebra, tridiagonal algebra, Alg<tex> ${\pounds}$</tex>