I’m wondering under what hypothesis it is true a property like

$$(mathcal{H}_1cap X, mathcal{H}_2cap X)_{theta}=mathcal{H}_1cap Xcap (mathcal{H}_1, mathcal{H_2})_{theta}$$

where $mathcal{H}_2hookrightarrow mathcal{H}_1$ are Hilbert spaces contained in a larger Hilbert space $mathcal{H}$ with $Xsubset mathcal{H}$.

I’m not skilled in interpolation theory, but here is my attemp. In Triebel’s book Section 1.17.1 (https://www.sciencedirect.com/bookseries/north-holland-mathematical-library/vol/18) there is a Theorem which read as follows

**Theorem 1:** *Let ${A_0, A_1}$ be an interpolation couple. Let $B$ be a complemented subspace of $A_0+A_1$ whose projection belongs to $L({A_0, A_1}, {A_0, A_1})$. Let $F$ be an arbitrary interpolation functor. Then ${A_0cap B, A_1cap B}$ is also an interpolation couple and*

$$F({A_0cap B, A_1cap B})=F({A_0, A_1})cap B$$

In my case, the interpolation couple would be ${mathcal{H}_2, mathcal{H}_1}$ and $B=mathcal{H}_1cap X$. If $X$ is such that $mathcal{H}_1cap X$ is a closed subspace of $mathcal{H_1}$ then it is also a complemented subspace of $mathcal{H}_1$ whose projection is linear continuous in $mathcal{H}_1$ (i.e. belongs to $L(mathcal{H}_1)$). The previous reasoning implies that I’m able to apply the previous Theorem to arrive my initial statement, or I’m missing something?

I know that interpolation is not well behaved with respect to restriction (https://math.stackexchange.com/questions/3542640/complex-interpolation-and-intersection) and I didn’t find much more results than the previous one in the literature. Every hint or reference is very well received!

**Remark:** I asked in some generality, but I’m treating a particular case where $mathcal{H_2}, mathcal{H_1}$ are sobolev spaces $H^k$, the larger space is $L^2$ and $X$ is the domain of a maximal monotone operator, in some interval $(0, L)$.