Unitary Interpolation problems in CSL-Algebra Alg$\Cal L$

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 $Ax_i=y_i$ for $i=1,2,\cdots,n$. In this article, we investigate unitary interpolation problems in CSL-Algebra Alg$\Cal L$ : Let $\Cal L$ be a commutative subspace lattice on a Hilbert space $\Cal H$. Let $x$ and $y$ be vectors in $\Cal H$. When does there exist a unitary operator $A$ in Alg$\Cal L$ such that $Ax=y$ ?