Let $w=\{w_k\}_{k\in \mathbb{N}}$ be a positive sequence and consider the weighted $\ell^p$-space defined by \begin{equation} \ell_w^p=\{c=\{c_k\}_{k\in \mathbb{N}}:\|\{c_kw_k\}_{k\in \mathbb{N}}\|_{\ell^p}<\infty\} \end{equation} and the associated norm \begin{equation} \|c\|_{\ell_w^p}=\|\{c_kw_k\}_{k\in \mathbb{N}}\|_{\ell^p}. \end{equation} Suppose that $T\in \mathcal{B}(\ell_w^1)$ and $T\in \mathcal{B}(\ell_w^{\infty})$ (where $\mathcal{B}(X)$ denotes the space of bounded operators on the Banach space $X$), then I would like to conclude that $T\in\mathcal{B}(\ell_w^p)$ for all $1<p<\infty$.
Remark: Note that $\ell_w^p$ is different from $L^p(\mathbb{N},w\cdot \mu)$ ($\mu$ denotes the counting measure) since the measure changes with $p$. Therefore the claim does not follow from the Riesz-Thorin Theorem. I was able to find a few theorems that deal with interpolation of $L^p$-spaces with change of measure but none of them implied the claim.
I am looking for a simple proof of the statement or a reference to a theorem in the literature that would imply my claim.
Consider the operator $S \colon \ell^\infty \to \ell^\infty_w$ given by $Sx = (x_n/w_n)$, note that $S$ is an isometric isomorphism, that likewise maps $\ell^p$ isometric onto $\ell^p_w$. Now $U = S^{-1}TS$ is bounded from $\ell^\infty$ into $\ell^\infty$ and form $\ell^1$ into $\ell^1$. Riesz-Thorin gives you that $U$ is bounded from $\ell^p$ into $\ell^p$. Hence $T = SUS^{-1}$ is bounded from $\ell^p_w$ into $\ell^p_w$.