Given vectors x and y in a separable complex Hilbert space $mathcal{H}$, an interpolating operator is a bounded operator A such that Ax = y. We show the following: Let $Alg{mathcal{L}}$ be a tridiagonal algebra on $mathcal{H}$ and let $x=(x_i)$ and $y=(y_i)$ be vectors in $mathcal{H}$. Then the following are equivalent: (1) There exists a unitary operator $A=(a_{ij})$ in $Alg{mathcal{L}}$ such that Ax = y. (2) There is a bounded sequence ${{alpha}_i}$ in $mathbb{C}$ such that ${mid}{alpha}_i{mid}=1$ and $y_i={alpha}_ix_i$ for $i{in}mathbb{N}$.