Suppose that $X_{\theta}$ is an interpolation space between the Banach spaces $X_0$ and $X_1$. Let $\mathcal{B}$ be another Banach space.

Is it true that $X_{\theta}\times\mathcal{B}$ is an interpolation space between $X_{0}\times\mathcal{B}$ and $X_{1}\times\mathcal{B}$?

Thank you for any suggestion.


1 Answer 1


Yes, interpolation on product spaces works componentwise, so $$\Bigl(\prod_{i=1}^n X_i,\prod_{i=1}^n Y_i\Bigr) = \prod_{i=1}^n (X_i,Y_i)$$ for any interpolation functor $(\cdot,\cdot)$ even with equal norms for a fixed choice of $\ell_p$ norm on the product spaces. This follows from the restriction/corestriction theorem in Section 1.2.4 in the bible book of Triebel (or alternatively from interpolation for complemented subspaces in Section 1.17.1): One shows that the $j$-th component of the interpolation space of products is exactly $(X_j,Y_j)$.

For this, let $R_j$ be the mapping which extracts the $j$-th component of an $n$-tuple, and let $E_j$ insert the $j$-th component in an $n$-vector of zeroes. Then $E_jR_j$ is a linear continuous projection and $E_jR_j\prod_{i=1}^n X_i$ is a complemented subspace which is isometrically isomorphic to $X_j$; analogously for $Y_j$. Hence, by the mentioned theorem(s), $$E_jR_j\Bigl(\prod_{i=1}^n X_i,\prod_{i=1}^n Y_i\Bigr) =(X_j,Y_j)$$ and this again isometrically.

Triebel, Hans, Interpolation theory. Function spaces. Differential operators, Berlin: Deutscher Verlag des Wissenschaften. 528 p. M 87.50 (1978). ZBL0387.46033.


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