Consider parameter $s$ in definition of $R(s,A)=(s I - A)^{-1}$ where $A$ is a linear operator in a vector space $X$. When $X$ is over $\mathbb{C}$, then $s$ is thought to be a complex number.
Now consider a very special case when $X$ is a Hilbert space and $A$ is a self-adjoint operator. There is a one-parameter family of unitary operators canonically associated with $A$: $ \pi(t) = \exp(i t A)= \int_{\mathbb{R}}{ \exp(i t x) d\mu_A(x)}$, $t\in\mathbb{R}$.
Here $\mu_A$ is a projector-valued measure supported at the spectrum $\sigma(A)$; $\pi(\cdot)$ satisfies a group property $\pi(t+s)=\pi(t)\pi(s)$ and is a strongly continuous unitary representation of $\mathbb{R}$. Any strongly continuous representations of $\mathbb{R}$ has this form by Stone's theorem + spectral theorem.
Furthermore, any strongly continuous unitary representation of locally compact Abelian group (LCA-group) $G$ can be decomposed into a direct sum of irreducible representations: $ \pi(g) = \int_{\hat{G}}{ \chi(g) d\mu(\chi)}, g\in G. $ Here $\mu$ is a projector-valued measure on Borel subsets of $\hat{G}$, Pontryagin dual to $G$. Support $\mathrm{supp}\mu$ can be considered as a spectrum of the representation $\sigma(\pi)$.
We cannot define a generating operator $A$ in the above context. But what about resolvent $R(\chi,\pi)$? Its parameter should take value in $\hat{G}$, Pontryagin dual to $G$, or, more generally, in a group of all quasicharacters $\chi:G\to\mathbb{C}^\times$. Ideally, resolvent should be defined for all $\chi\not\in\sigma(\pi)$.
The situation with resolvent $R(\chi,\pi)$ is not hopeless. In the case of contraction operator semigroups described by Hille--Yoshida theorem we can compute the resolvent directly for some values of $s$ without mentioning generator: $ R(s,A) = \int_{[0,+\infty)}{ \exp(- s t) \pi(t) dt}.$ Here $\pi(t)=\exp(t A)$ form a contraction operator semigroup, $t\in[0,+\infty)$; $A$ is a generator with spectrum in the left part of the complex plane.
In any LCA-group $G$ we can consider a closed semigroup $G_+\in G$ (of non-zero Haar measure), an operator semigroup $\pi(g), g\in G_+$, and a quasicharacter $\chi$. We can define a ``resolvent'' $R(\chi,\pi)= \int_{G_+}{ \chi(t) \pi(t) dt }$.
1) Is there a notion of a ``resolvent of a representation'' defined somewhere in the literature?
2) Is the construction with integral over semigroup studied somewhere for semigroups other then $[0,+\infty)$?
3) Can some analogue of Hilbert equality $R(s,A)-R(t,A) = - (s-t) R(s,A) R(t,A)$ be introduced for the ``resolvent'' $R(\chi,\pi)$ above?
