Let $\mathscr{H}$ be a separable complex Hilbert space and $H$ be an (unbounded) self-adjoint operator on $\mathscr{H}$ bounded from below, e.g., $H\ge 0$. Suppose that $e^{-H}$ is of trace-class so that it is diagonalized by the orthonormal basis $\varphi_n$ with eigenvalues $\lambda_n$. Then is it true that $\varphi_n$ diagonalizes $H$ so that $H\varphi_n = -\varphi_n \log \lambda_n$?
I think I can prove that $0<\lambda_n \le 1$ for all $n$ so that $-\log \lambda_n \ge 0$ is well-defined, but how do we know that $\varphi_n$ are in the domain of $H$?