A linear operator bounded in all Besov but not in Triebel-Lizorkin spaces

We construct a linear operator $T:{\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^n)\to {\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^n)$ such that $T:{\mathcal{B}}_{pq}^s({\mathbb{R}}^n)\to {\mathcal{B}}_{pq}^s({\mathbb{R}}^n)$ for all $0<p,q\le \mathrm{\infty }$ and $s\in \mathbb{R}$, but $T({\mathcal{F}}_{pq}^s({\mathbb{R}}^n))\subset ̸{\mathcal{F}}_{pq}^s({\mathbb{R}}^n)$ unless $p=q$. As a result Triebel-Lizorkin spaces cannot be interpolated from Besov spaces unless $p=q$. In the appendix we purpose a question for the interpolation framework via structured Banach spaces.