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$\mathfrak{L}_{II}$ operator is introduced by Y.-L. Xin (\emph{Calculus ofVariations and Partial Differential Equations. 2015,\textbf{54}(2):1995-2016)}, which is an important extrinsic ellipticdifferential operator of divergence type and has profound geometric meaning. Inthis paper, we extend $\mathfrak{L}_{II}$ operator to more general ellipticdifferential operator $\mathfrak{L}_{\nu}$, and investigate the clamped plateproblem of bi-$\mathfrak{L}_{\nu}$ operator, which is denoted by$\mathfrak{L}_{\nu}^{2}$, on the complete Riemannian manifolds. A generalformula of eigenvalues for the $\mathfrak{L}_{\nu}^{2}$ operator isestablished. Applying this formula, we estimate the eigenvalues with lowerorder on the Riemannian manifolds. As some further applications, we establishsome eigenvalue inequalities for this operator on the translating solitons withrespect to the mean curvature flows, submanifolds of the Euclidean spaces, unitspheres and projective spaces. In particular, for the case of translatingsolitons, all of the eigenvalue inequalities are universal.

Eigenvalue Inequalities for the Clamped Plate Problem of $\mathfrak{L}^{2}_{\nu}$ Operator