Question

Riesz-Thorin interpolation may sometimes be applied to subspaces (of $\ell^p$ or $L^p$) when these are complemented and the spaces in the complementation comes from a common dense subspace. To be a bit more formal, if there exists $A$ and $B$ such that if $A_p = \overline{A}^{\ell^p}$ and $B_p = \overline{B}^{\ell^p}$, and if $\forall q \in [q_1,q_2]$, $A_q \oplus B_q = \ell^q$, then the interpolation may be done between values of $q$ in this interval.

I was wondering what techniques are available to show this situation holds...? To make the question more precise, here is a case (which may be constructed):

$\textbf{Question}$: fix some $K>1$ and assume that two subspaces $A_p$ and $B_p$ of $\ell^p(\mathbb{N})$ are given by the closure of the span of $\lbrace a_i \rbrace_{i \in \mathbb{N}}$ and $\lbrace b_i \rbrace_{i \in \mathbb{N}}$ respectively. These spanning elements are combinatorially not too messy: the $a_i$ and $b_i$ may be written as the a difference of characteristic functions $\chi_S - \chi_T$ where $S$ and $T$ are finite sets in $\mathbb{N}$ of cardinality less than $K$. Furthermore, for a given $n \in \mathbb{N}$, the number of $i$ such that $a_i$ or $b_i$ is non zero at $n$ is less than $K$. Finally, $A_2$ is actually the (orthogonal) complement of $B_2$ in $\ell^2(\mathbb{N})$. Are there $p \in [1,\infty]$ (and $p \neq 2$), for which one may conclude that $A_p + B_p = \ell^p(\mathbb{N})$?

(For $p>2$ one has obviously that $\overline{A_p + B_p} = \ell^p(\mathbb{N})$.)

The description of the span of the spaces avoid the examples known to me of uncomplemented subspaces ($1 \leq p<\infty$), but it may very well not exclude all possibilities. Apologies in advance if this makes an easy negative answer.

EDIT: I realized the answer may not be positive for $p$ "too small" (where "too small" depends perhaps on $K$) as one would obtain contradictions with some known results.

Answer 1

You do get complementation in the case you mention. Two key facts that you did not state explicitly but which follow easily from your hypotheses are that every unit vector $e_n$ is in the linear span of at most $K$ of the $a_i$ and $b_i$, and the span of $K$ of the $a_i$ and $b_i$ is contained in the span of at most $N=2K^2$ unit vectors. Using these facts, it is not hard to verify that the orthogonal projection $P$ onto $A$ is bounded in the $\ell_1$, which by interpolation and duality gives you what you want. To check boundedness in $\ell_1$, you just have to give a bound on $\|Pe_n\|_1$ that is independent of $n$. But on the span of $N$ unit vectors, the $\ell_1$ norm is dominate by $N^{1/2}$ times the $\ell_2$ norm.

EDIT: As Antoine points outs, my answer is wrong.