Let $A: \ell^2 \rightarrow \ell^2$ be a bounded operator given by \begin{equation} (Au)(\alpha) = \sum_{\beta}A(\alpha,\beta)u(\beta) \end{equation} where $\left|A(\alpha,\beta) \right|\le Ce^{-|\alpha-\beta|}.$
Now assume that $B=UAU^*$ and $U$ is unitary on $\ell^2$ with $(Bu)(\alpha) = \sum_{\beta}B(\alpha,\beta)u(\beta).$
I would like to know if there is also exponential decay of the coefficients $B(\alpha,\beta)$ under these assumptions?
Consider $\ell^2({\mathbb N_0})$. Let $A(\alpha,\beta)=e^{-|\alpha-\beta|}$. Let $\sigma$ be a permutation of ${\mathbb N}_0$ such that $\sigma(0)=0$. Let $U(u)(\alpha)=u(\sigma^{-1}\,(\alpha))$. Then the estimate $|B(\alpha,\beta)|\le Ce^{|\alpha-\beta|}$ gives for $\beta=0$ that $e^{-\sigma^{-1}(\beta)}\le Ce^{-\beta}$, or $$ e^{\sigma(\beta)-\beta}\ \ \ \le C $$ for every natural number $\beta$, so the permutation has to have bounded distance between $\beta$ and $\sigma(\beta)$. It is easy to construct e permutation which doesn't have bounded distance.
