I have a linear operator $O$ acting in real vector space $\mathbb{R}^n$. It turns out that the eigenvector $\vec{v}^*$, corresponding to smallest eigenvalue (potentially negative) $\lambda^*$ of the matrix of $O$ in $\mathbb{R}^n$ (in a given orthonormal basis), is mostly spanned by vectors from some subspace $\mathbb{R}^m\subset \mathbb{R}^n$ of much smaller dimension $m\ll n$. By "mostly spanned" I mean that the absolute values of components of $\vec{v}^*$ corresponding to $\mathbb{R}^m$ are much smaller than those corresponding to its orthogonal complement:
$$ \vec{v}^* = (\underbrace{v_1, v_2, \ldots, v_m}_{\mathbb{R}^{m}},\underbrace{v_{m+1}, v_{m+2}, \ldots, v_n}_{\mathbb{R}^{n-m}}) \, ,\\ |v_i| \ll |v_j| \; \text{for} \; i\in[1,m],\,j\in[m+1,n]\, . $$
Under which conditions can one use the restriction of $O$ to the $\mathbb{R}^m$ to estimate $\vec{v}^*$? I.e., to say that the eigenvector, corresponding to the lowest eigenvalue of the matrix of $O$ in $\mathbb{R}^m$, is approximately equal to $\vec{v}^*$.