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 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.

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.