# Complemented Subspaces and Riesz-Thorin interpolation

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.

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.
• Apologies for being slow, but there are a few points I am definitively missing. Take a particularly singular example (where $B$ is trivial) with $a_i = e_i + e_{i-1}$ if $i>1$ and $a_1 = e_1$. Then $A_2 = \ell_2$ and it seems that $e_n$ is in the span of $n$ of the $a_i$ (i.e. not uniformly bounded by $K$), though $K=3$ suffices. I can cook up a more interesting case (with $B$ non-trivial) if it is deemed interesting.