Let $(h,D(h))$ be a self-adjoint operator on the infinite dimensional Hilbert space $\mathfrak{h}$ that is bounded below and also assume that tr$_{\mathfrak{h}}(e^{-h})<\infty$.
How can I see that $h$ has purely discrete spectrum and that the sequence of eigenvalues $e_0\leq e_1\leq e_2\leq...$ repeated according to multiplicity is unbounded? And moreover why is tr$_{\mathfrak{h}}(e^{-\beta(h-\mu)})<\infty$ for $\beta>0$ and $\mu\in\mathbb{R}$?