Is there an elliptic operator $D$ on $C^{\infty}(S^2)$ whose principal symbol is not identical to thats of Laplacian but it satisfies $\int_{S^2} fDf =\int_{S^2} f\Delta (f)$ for all $f\in C^{\infty}(S^2)$?
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No: the operator must be of even order, and therefore the symbol of the adjoint operator $D^*$ is the same as the symbol of $D$. Hence we can assume without loss of generality that $D$ is symmetric. By applying the functional identity to the family of functions $f+t g$ for $t\in(-\epsilon,\epsilon)$ one easily finds that $$\int_M fDg\, dvol=\int_Mf\Delta g\, dvol$$ for arbitrary functions $f,g$. Hence $D=\Delta$.
