# Discrete Lions Peetre interpolation

In S. Heinrich's "Closed operator ideals and interpolation," Heinrich describes two equivalent descriptions of Lions-Peetre interpolation space $(X,Y)_{\theta, p}$ for $0<\theta<1$ and $1\leqslant p<\infty$. We first assume that $X,Y$ are Banach spaces which are both included in an ambient space $Z$. We then define the norms $$\|x\|_{X\cap Y}= \max\{\|x\|_X, \|y\|_Y\}$$ and $$\|z\|_{X+Y}=\inf\{\|x\|_X+\|y\|_Y: x\in X, y\in Y, z=x+y\}.$$

For two numbers $\xi_1, \xi_2$ such that $\xi_1(\xi_1-\xi_2)^{-1}=\theta$, $(X,Y)_{\theta, p}$ is (up to equivalent norms) the space of $y\in X+Y$ such that we may write $y=\sum_{n=-\infty}^\infty y_n$ with $y_n \in X\cap Y$, $\sum_{n=-\infty}^\infty \|e^{\xi_1 n}y_n\|_X^p, \sum_{n=-\infty}^\infty \|e^{\xi_2 n}y_n\|_Y^p<\infty$ endowed with the norm $|\cdot|$ such that for $y\in (X,Y)_{\theta, p}$, $|y|$ is given by $$\Bigg\{\max\Bigl\{\bigl(\sum_{n=-\infty}^\infty \|e^{\xi_1 n}y_n\|^p_X\bigr)^{1/p}, \bigl(\sum_{n=-\infty}^\infty \|e^{\xi_2 n}y_n\|^p_Y\bigr)^{1/p} \Bigr\}: y=\sum_{n=-\infty}^\infty y_n, y_n\in X\cap Y\Biggr\}.$$ This space is also equal to (up to equivalent norms) the space of all $y\in X+Y$ such that for each $n=0, \pm 1, \ldots$, we may decompose $y=y_{1n}+y_{2n}$, $y_{1n}\in X$, $y_{2n}\in Y$ so that $\sum_{n=0}^\infty \|e^{\xi_1 n}x_{1n}\|_X^p, \sum_{n=0}^\infty \|e^{\xi_2 n}x_{2n}\|_Y^p<\infty$ endowed with the norm $[y]$ on $(X, Y)_{\theta, p}$ by $$[y]=\inf \Bigg\{ \bigl(\sum_{n=0}^\infty \|e^{\alpha n}x_{1n}\|_X^p\bigr)^{1/p}, \bigl(\sum_{n=0}^\infty \|e^{\xi_2 n} x_{2n}\|_Y^p\bigr)^{1/p}: y=y_{1n}+y_{2n}, y_{1n}\in X, y_{2n}\in Y\Biggr\}$$

In this case, if $E_n$ is $X\cap Y$ endowed with the norm $$\|x\|_{E_n}=\max\{\|e^{\xi_1 n} x\|_X, \|e^{\xi_2 n} x\|_Y\}$$ and if $F_n$ is $X+Y$ endowed with the norm $$\|y\|_{F_n}= \inf \{\|e^{\xi_1 n}y_1\|_X+ \|e^{\xi_2 n}y_2\|_Y: y=y_1+y_2, y_1\in X, y_2\in Y\},$$ then there exist a canonical surjection $Q:(\oplus_{n=-\infty}^\infty E_n)_{\ell_p(\mathbb{Z})}\to (X,Y)_{\theta, p}$ given by $Q((y_n)_{n=-\infty}^\infty)=\sum_{-\infty}^\infty y_n$ such that $|y|=\inf \{\|z\|: Qz=y\}$ and a canonical isomorphic emedding $J:(X, Y)_{\theta, p}\to (\oplus_{n=-\infty}^\infty F_n)_{\theta, p}$ given by $J(y)=(\ldots, y, y, \ldots)$. Furthermore, if $P_m:(\oplus_{n=-\infty}^\infty E_n)_{\ell_p(\mathbb{Z})}\to E_m$, $Q_m:(\oplus_{n=-\infty}^\infty F_j)_{\ell_(\mathbb{Z})}\to F_m$ are the canonical projection, then $Q_mJQP_n$ is the canonical inclusion of $X\cap Y$ into $X+Y$ up to equivalent norms on the domain and range.

Does the analogous construction hold for a $c_0$ sum? More precisely, if we define $U$ to be the space of all $y\in X+Y$ such that we may write $y=\sum_{n=-\infty}^\infty y_n$, $y_n\in X\cap Y$ and $(\|e^{\xi_1 n}y_n\|_X)_{n=-\infty}^\infty, (e^{\xi_2 n}y_n)_{n=-\infty}^\infty \in c_0(\mathbb{Z})$ with the norm $|\cdot|$ given by $$|y|= \Bigl\{ \Bigl\|(\|e^{\xi_1 n}y_n\|_X)_{n=-\infty}^\infty\Bigr\|_{c_0(\mathbb{Z})},\Bigl\|(\|e^{\xi_2 n} y_n\|_Y)_{n=-\infty}^\infty\Bigr\|_{c_0(\mathbb{Z})}: y=\sum_{n=-\infty}^\infty y_n, y_n\in X\cap Y\Bigr\}$$ and $V$ to be the space of all $y\in X+Y$ such that we may decompose $y=y_{1n}+y_{2n}$, $y_{1n}\in X$, $y_{2n}\in Y$ such that $(\|e^{\xi_1 n}y_{1n}\|_X)_{n=-\infty}^\infty, (\|e^{\xi_2, n}y_{2n}\|_Y)_{n=-\infty}^\infty\in c_0(\mathbb{Z})$ endowed with the norm $$[y]=\inf \Bigl\{\Bigl\|\bigl(\|e^{\xi_1 n}y_{1n}\|_X\bigr)_{n=-\infty}^\infty\Bigr\|_{c_0(\mathbb{Z})}\vee \Bigl\|\bigl(\|e^{\xi_2 n}y_{2n}\|_Y \bigr)_{n=-\infty}^\infty\Bigr\|_{c_0(\mathbb{Z})}\Bigr\},$$ where the infimum is taken over all appropriate decompositions. Then are $U$ and $V$ equal as sets with equivalence of $|\cdot|$ and $[\cdot]$, and do there exist a surjection $Q:(\oplus_{n=-\infty}^\infty E_n)_{c_0(\mathbb{Z})}\to U$ and isomorphic embedding $J:V\to (\oplus_{n=-\infty}^\infty F_n)_{c_0(\mathbb{Z})}$?