Given vectors x and y in a separable complex Hilbert space $cal{H}$, an interpolating operator is a bounded operator A such that Ax = y. We show the following : Let Alg$cal{L}$ be a tridiagonal algebra on a separable complex Hilbert space H and let x = ($x_i$) and y = ($y_i$) be vectors in H. Then the following are equivalent: (1) There exists an invertible operator A = ($a_{kj}$) in Alg$cal{L}$ such that Ax = y. (2) There exist bounded sequences ${{alpha}_n}$ and ${{{eta}}_n}$ in $mathbb{C}$ such that for all $kinmathbb{N}$, ${alpha}_{2k-1} eq0,;{eta}_{2k-1}=frac{1}{{alpha}_{2k-1}},;{eta}_{2k}=frac{alpha_{2k}}{{alpha}_{2k-1}alpha_{2k+1}}$ and $$y_1={alpha}_1x_1+{alpha}_2x_2$$ $$y_{2k}={alpha}_{4k-1}x_{2k}$$ $$y_{2k+1}={alpha}_{4k}x_{2k}+{alpha}_{4k+1}x_{2k+1}+{alpha}_{4k+2}x_{2k+2}$$.