  Let $M$ be an $n(>2)$-dimensional closed orientable submanifold in an
$(n+p)$-dimensional space form $\mathbb{R}^{n+p}(c)$. We obtain an optimal
upper bound for the second eigenvalue of a class of elliptic operators on $M$
defined by $L_{T}f=-div(T\nabla f)$, where $T$ is a general symmetric, positive
definite and divergence-free $(1,1)$-tensor on $M$. The upper bound is given in
terms of an integration involving $tr T$ and $|H_T|^2$, where $tr T$ is the
trace of the tensor $T$ and $H_T=\sum_{i=1}^nA(Te_i,e_i)$ is a normal vector
field associated with $T$ and the second fundamental form $A$ of $M$.
Furthermore, we give the sufficient and necessary conditions when the upper
bound is attained. Our main theorem can be viewed as an extension of the famous
`Reilly inequality'. The operator $L_{T}$ can be regarded as a natural
generalization of the well-known operator $L_r$ which is the linearized
operator of the first variation of the $(r+1)$-th mean curvature for
hypersurfaces in a space form. As applications of our main theorem, we
generalize the results of Grosjean ([16]) and Li-Wang ([19]) for hypersurfaces
to higher codimension case.
