Let $(M,g)$ be a closed Riemannian manifold, $(\lambda_n)_{n=1}^{\infty}$ be a square-summable monotonically decreasing sequence of non-negative numbers, and let $(f_k)_{k=1}^{\infty}$ be an orthonormal basis in $L^2(M)$ (defined wrt. the Riemannian volume measure on $(M,g)$).

Does there exist/can we construct a second-order elliptic operator $T$ on $M$ with smooth coefficients?