SELF-ADJOINT INTERPOLATION ON AX=Y IN A TRIDIAGONAL ALGEBRA ALG𝓛

Abstract

Given operators X and Y acting on a separable Hilbert space <TEX>$\mathcal{H}$</TEX>, an interpolating operator is a bounded operator A such that AX = Y. In this article, we investigate self-adjoint interpolation problems for operators in a tridiagonal algebra : Let <TEX>$\mathcal{L}$</TEX> be a subspace lattice acting on a separable complex Hilbert space <TEX>$\mathcal{H}$</TEX> and let X = (<TEX>$x_{ij}$</TEX>) and Y = (<TEX>$y_{ij}$</TEX>) be operators acting on <TEX>$\mathcal{H}$</TEX>. Then the following are equivalent: (1) There exists a self-adjoint operator A = (<TEX>$a_{ij}$</TEX>) in <TEX>$Alg{\mathcal{L}}$</TEX> such that AX = Y. (2) There is a bounded real sequence {<TEX>${\alpha}_n$</TEX>} such that <TEX>$y_{ij}={\alpha}_ix_{ij}$</TEX> for <TEX>$i,j{\in}\mathbb{N}$</TEX>.