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 $Tx_i;:;y_i,;for;i;=;1,;2,;{cdots},;n$. In this article, we obtained the following : $Let;x;=;{x_i};and;y={y_}$ be two vectors in a separable complex Hilbert space H such that $x_i; eq;0$ for all $i;=;1,;2;cdots$. Let L be a commutative subspace lattice on H. Then the following statements are equivalent. (1) $sup;{frac{$mid${sum_{k=1}}^l;alpha_{kappa}E_{kappa}y$mid$}{$mid${sum_{k=1}}^l;alpha_{kappa}E_{kappa}x$mid$};:;l;in;mathbb{N},;alpha_{kappa};in;mathbb{C};and;E_{kappa};in;L};<;infty;and;$mid$y_n$mid$x_n$mid$^{-1};=;1;for;all;n;=;1,;2,;cdots$</TEX>. (2) There exists an operator A in AlgL such that Ax = y, A is a unitary operator and every E in L reduces, A, where AlgL is a tridiagonal algebra.