Given a self-adjoint operator $\hat{T}$ on a Hilbert space $\mathcal{H}$, and assuming it has a basis of eigenvectors $\{\phi_n\}$ such that $\hat{T}\phi_n=\lambda_n\phi_n$, one can consider the subspace $$V=\textrm{span}\{\phi_n:\lambda_n>0\},$$ inside of which $\hat{T}$ is positive definite. My question is, does there exist a way of defining this subspace invariantly in terms of $\hat{T}$ without making reference to its eigenvectors?

(Some motivation: given the hydrogen-atom hamiltonian in quantum mechanics, $\hat{H}=-\frac{\hbar^2}{2m}\nabla^2 -\frac{e^2}{r}$, the subspace where $\hat{H}<0$ is the linear span of all bound states, which correspond to elliptical orbits in the analogous classical problem.)