Let $H$ be a $\mathbb R$-Hilbertspace, $A$ be a self-adjoint linear contraction on $H$ and $E:\mathcal B(\mathbb R)\to\mathfrak L(H)$ denote the spectral measure associated with $A$. By contractivity, the spectrum $\sigma(A)$ of $A$ is contained in the closed unit ball $\overline B_1(0)$ around $0$ and hence $$A^n=\int_{B_1(0)}\lambda^n\:E({\rm d}\lambda)+\int_{\partial B_1(0)}\lambda^n\:E({\rm d}\lambda)\;\;\;\text{for all }n\in\mathbb N\tag1.$$
In the complex setting, we can show that (see Lemma 1 in this paper) $$A^n=\int_0^{2\pi}e^{{\rm i}\lambda n}\left(\left.E\right|_{\partial B_1(0)}+f(\lambda){\rm d}\lambda\right),\tag2$$ where $$f(\lambda):=\frac1{2\pi}\int_{B_1(0)}\frac{1-|\mu|^2}{|1-\mu e^{{\rm i}\lambda}|^2}\:E({\rm d}\mu)\;\;\;\text{for }\lambda\in\mathbb R,$$ for all $n\in\mathbb N$. Is there a similar formula on the real case?