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

  • Received : 2013.11.20
  • Accepted : 2014.01.20
  • Published : 2014.03.25

Abstract

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

Keywords

self-adjoint interpolation;CSL-algebra;tridiagonal algebra;Alg$\mathcal{L}$

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