Given operators $X_i$ and $Y_i$ (i = 1, 2, ${cdots}$, n) acting on a Hilbert space $mathcal{H}$, an interpolating operator is a bounded operator A acting on $mathcal{H}$ such that $AX_i$ = $Y_i$ for i= 1, 2, ${cdots}$, n. In this article, if the range of $X_k$ is dense in H for a certain k in {1, 2, ${cdots}$, n), then the following are equivalent: (1) There exists a self-adjoint operator A in $mathcal{B}(mathcal{H})$ stich that $AX_i$ = $Y_i$ for I = 1, 2, ${cdots}$, n. (2) $sup{{frac{{parallel}{sum}^n_{i=1}Y_if_i{parallel}}{{parallel}{sum}^n_{i=1}X_if_i{parallel}}:f_i{in}H}}$ < ${infty}$ and < $X_kf,Y_kg$ >=< $Y_kf,X_kg$> for all f, g in $mathcal{H}$.