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

$$[\mathcal{H}_1\cap X, \mathcal{H}_2\cap X]_{\theta}=\mathcal{H}_1\cap X\cap [\mathcal{H}_1, \mathcal{H_2}]_{\theta}$$

where $\mathcal{H}_2\hookrightarrow \mathcal{H}_1$ are Hilbert spaces contained in a larger Hilbert space $\mathcal{H}$ with $X\subset \mathcal{H}$.

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 $\{A_0\cap B, A_1\cap B\}$ is also an interpolation couple and $$F(\{A_0\cap B, A_1\cap B\})=F(\{A_0, A_1\})\cap B$$

EDIT: The space $L(\{A_0, A_1\}, \{B_0, B_1\})$ denotes the set of all linear operators mapping $A_0+A_1$ into $B_0+B_1$ such that their restrictions to $A_k$, $k=0$, $1$ are continuous mappings from $A_k$ into $B_k$.

In my case, the interpolation couple would be $\{\mathcal{H}_2, \mathcal{H}_1\}$ and $B=\mathcal{H}_1\cap X$. If $X$ is such that $\mathcal{H}_1\cap X$ 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 $(0, L)$.

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    $\begingroup$ I think that what your are missing is that you need that $H_1 \cap X$ and $H_2 \cap X$ are complemented subspaces of $H_1$ and $H_2$ with compatible projections. $\endgroup$
    – Mikael de la Salle
    Commented Feb 6, 2021 at 22:20
  • $\begingroup$ @MikaeldelaSalle Sorry, but what do you mean by compatible projections? Although "compatible" is a widely used concept, I don't know it in this context and google doesn't seem to help me. Thanks! $\endgroup$
    – rebo79
    Commented Feb 6, 2021 at 22:47
  • $\begingroup$ Thinking about it, you mean that if $P: \mathcal{H}_1\to \mathcal{H}_1\cap X$ is the (linear continuous) projector, then $P$ when restricted to $\mathcal{H}_2$ is the (linear continuous) projector to $\mathcal{H}_2\cap X$, right? $\endgroup$
    – rebo79
    Commented Feb 6, 2021 at 23:06
  • $\begingroup$ Yes, that us what meant by compatible, and (I Guess) what Triebel means by $L(\{A0,A1\},\{A0,A1\})$. Note however that the projections need not be orthogonal. $\endgroup$
    – Mikael de la Salle
    Commented Feb 7, 2021 at 13:38
  • $\begingroup$ @MikaeldelaSalle I edited the post with the definition given by Triebel, and indeed is what you mean. At the very end in my particular case $\mathcal{H}_2$ is not closed in $\mathcal{H}_1$ (is a dense subspace instead), so I cannot expect that $P$ is still continuous when restricted to $\mathcal{H}_2$. I'm grateful with your remarks, thanks! $\endgroup$
    – rebo79
    Commented Feb 7, 2021 at 19:13

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