Intersection of the kernel with the interpolation space

Question

$\DeclareMathOperator\Ker{Ker}$Given two Banach spaces $X$ and $Y$ with a continuous inclusion $X\subset Y$, and another couple $X’ \subset Y’$ with the same properties. Take $f : Y \longrightarrow Y’$ linear continuous, such that $f_{\mid X}$ induces a linear continuous map from $X$ to $X'$. My question is for $0<s<1$ and $p>1$, is the following true:

$$(X\cap \Ker(f),\Ker(f))_{s,p}=\Ker(f) \cap (X,Y)_{s,p}\;\;?$$

Here I'm considering the K-method for the interpolation.

The inclusion $(X\cap \Ker(f),\Ker(f))_{s,p}\subset \Ker(f) \cap (X,Y)_{s,p}$ follows directly from the definition. My problem is the other inclusion.

It's clear that if we have $Z\subset Y$ then in general the following is not true:

$$(X\cap Z,Z)_{s,p}=Z \cap (X,Y)_{s,p},$$

one can take $X=H^{2}(U)$, $Z=H^{1}(U)$, $Y=L^{2}(U)$, $s=\frac{1}{2}$, and $p=2$. But this does not contradict our case (because $Z$ here is not even closed in $Y$).

If what I'm asking is not true in general, is it true under the following assumptions:

• $f(X)$ is closed in $X'$, and $f(Y)$ is closed in $Y'$, and

• $f$ is open onto $f(Y)$, and $f_{\mid X}$ is open onto $f(X)$.

Answer 1

(Making CW as this is an extended comment.)

Let's generalize slightly¹: let $X\hookrightarrow Y$ be an embedding of Banach spaces, and let $S\subset Y$ be a closed subspace with the induced norm.

Your question essentially boils down to comparing the $K$ functionals $K$ and $K^{(S)}$, where $$ K(t,z) = \inf_{z = x + y} \|x\|_X + \|y\|_Y $$ and $K^{(S)}$ is analogously defined, except that instead of allowing arbitrary decompositions $z = x+y$ with $x\in X$ and $y\in Y$, you are looking at decompositions with $x \in X\cap S$ and $y\in S$. (Obviously here $z\in S$ as well.)

Clearly $K(t,z) \leq K^{(S)}(t,z)$ for $z\in S$.

Your question will have a positive answer if $K(t,z) \gtrsim K^{(S)}(t,z)$ for some implicit constant independent of $t$ and $z$.

(This is why I objected to your "$H^1$ counterexample; the $K$-functional for the $(X\cap Z,Z)$ interpolation has almost nothing to do with the functional for the $(X,Y)$ interpolation in that case. Compared to this case where it is essentially a geometry question about how $S$ is situated in $X$ and in $Y$.)

A sufficient condition is therefore that there exists a projection $\Pi:Y\to S$ with the property that $\Pi(X) \subseteq X\cap S$ and the operator norms $\|\Pi\|_{Y\to S}$ and $\|\Pi\|_{X\to X\cap S}$ are both bounded.

(The existence of such projection operators is in general a non-trivial assumption, as for Banach spaces there are generally closed subspaces which are not images of continuous projections, and for Hilbert spaces non-orthogonal projections may fail to be bounded.²)

This would be the case if, for example, $X, Y$ were Hilbert, and $S$ is such that whenever $x\in X$ is orthogonal to $S$ w.r.t. the $X$ inner product, it is also orthogonal w.r.t. the $Y$ inner product.

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As Hannes noted in a comment below: the result alluded to above is given as Theorem 1 in Section 1.17.1 of Interpolation Theory, Function Spaces, Differential Operators (ed. Hans Triebel, North-Holland (1978)).

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¹: When $Y$ is separable, every closed subspace is a kernel, so in this case it is not more general. https://arxiv.org/abs/1811.02399

²: I don't see how to relate the existence of these projections to the additional conditions you are willing to assume, as stated in the question. But chances are this is because I don't know very much about complemented subspaces.