Let $H$ be a Hilbert space. For all $t\in\left]0,\infty\right[$ the operator $A_t\in L(H,H)$ is assumed to be trace-class and symmetric. Furthermore $A$ is a semigroup, i.e. $$A_{t+h}=A_tA_h$$ for all positive real numbers $t$ and $h$. I expect that we can prove the following theorem:
There is a countable set $M\subset\mathbb R$ and a family of subspaces $(V_\lambda)_{\lambda\in M}$ such that $$\forall t: A_t=\sum_{\lambda\in M}\exp(-t\lambda)P_\lambda$$ where $P_\lambda$ is the orthogonal projector onto $V_\lambda$.
I have an incomplete and unelegant proof using a basis (see below), can someone confirm the theorem/provide a reference?
Motivation: I think that this formula is implicitly used in the proof of the McKean-Singer formula in Heat Kernels and Dirac Operators (theorem $3.50$). For those who have the book, I think that we use $\mathrm{Str}(P_\lambda)=n^+_\lambda-n^-_\lambda$.
Addendum (proof based on the last paragraph of page $90$): Firstly, according to this answer we can simultaneously diagonalize the entire semi-group to obtain a Hilbert-basis $(v_n)$: $$A_tv_n=\mu_{n,t}v_n$$ The semi-group property implies that $\mu_{n,t}=\exp(-t\lambda_n)$ for some $\lambda_n$ (see this question). Hence, $$A_t=\sum_n\exp(-t\lambda_n)|n\rangle\langle n|$$ Assuming that $$P_\lambda=\sum_{n:\lambda_n=\lambda}|n\rangle\langle n|$$ and setting $M:=\{\lambda_n:n\in\mathbb N\}$ we should obtain the desired result, but I am not so sure about this step. (I asked a related question here.)
