If $(T^*)^n$ converges pointwise in $\ell^1$, what can we conclude about $T^n$?

Question

Suppose $T$ is a bounded linear operator on all $\ell^p$ spaces including $\ell^\infty$. Suppose I know that for all $x \in \ell^1$, $\lim_{n \to \infty} \| (T^*)^n x \|_{\ell^1}=0$, but the convergence is not uniform. Here $T^*$ is the formal adjoint of $T$ (if $T$ has matrix representation $T_{ij}=t_{ij}$ then $T^*$ has matrix representation $(T^*)_{ij}=t_{ji}$). Can I conclude anything useful about $T^n$? In particular I would like to have $\lim_{n \to \infty} \| T^n x \|_{\ell^\infty}=0$ for all $x \in \ell^\infty$ but I suspect that that is too optimistic.

Answer 1

The usual example with the right and left shifts apply. Let $$ Tx=(0,x_1,x_2,x_3,\ldots). $$ Then $Te_j=e_{j+1}$, so $T_{kj}=\delta_{k,j+1}$. Then $(T^*)_{kj}=\delta_{k,j-1}$, and $$ T^*x=(x_2,x_3,\ldots). $$ We have $$ \|(T^*)^nx\|_1=\sum_{k\geq n}|x_k|\to0, $$ while $$ \|T^nx\|_\infty=\|x\|_\infty. $$ The same $T$ gives a counterexample for any pair $p,q$ (not necessarily conjugate): $$ \|(T^*)^nx\|_p\to0,\ \ \|T^nx\|_q=\|x\|_q. $$