I am working through a text on Numerics for SPDEs and there the concept an interpolation (Hilbert-)space associated to an operator is used. To be specific:

Definition.Let $H$ be an $\mathbb{R}$-Hilbert space and $A: D(A) \rightarrow H$ a diagonal linear operator on $H$ with point spectrum $\sigma_P(A) \subseteq (0,\infty)$ and $\inf \sigma_P(A)>0$. Then the family $(H_r)_{r\in \mathbb{R}}$ of $\mathbb{R}$-Hilbert spaces with the properties

$\forall$ $r\geq s$ : $H_r \subset H_s = \overline{H_r}^{H_s}$

$\forall$ $r\in [0,\infty)$ : $(H_r, \langle \cdot,\cdot\rangle_{H_r}) = (D(A^r), \langle A^r(\cdot), A^r(\cdot)\rangle_H)$

$\forall$ $r\in (-\infty,0]$,$v\in H$ : $\|v\|_{H_r} = \|A^r v\|_{H}$

is called a family of interpolation spaces associated to $A$.

Now, I am searching for a concrete explanation of how this concept is useful in the analysis of SPDEs. By looking through the literature I found some hints, but no concise answer:

- It seems the interpolation spaces play a role in dealing with the nonlinearity F of the SPDE $dX_t = [AX_t+F(X_t)] dt+ B(X_t)dW_t$.
- They are somewhat of an analogue to Sobolev spaces and reveil a certain kind of regularity.

I would be very greatful if somebody has a good explanation/motivation of this idea.

Thanks in advance!