a linear map on $W$

Let $e_i\in W$ satisfy $(e_i)_j=1$ if $i=j$ and $(e_i)_j=0$ otherwise. Then $$\langle Te_i,e_j\rangle=\cases{1\,\,{\rm if}\,\,j\leq i,\\0 \,\,{\rm if}\,\,j>i.}$$

Thus if $T^*$ exists: $$\langle e_i,T^*e_j\rangle=\cases{1\,\,{\rm if}\,\,j\leq i,\\0 \,\,{\rm if}\,\,j>i.}$$

So we have $(T^*e_j)_i=1$ for all $i\geq j$ which contradicts $T^*e_j\in W$.

Suppose that $T^*$ exists. Exchanging the sum (no issues because every sequence is finite), $$ \langle T(a),b\rangle = \sum_{i=1}^\infty (\sum_{j=i}^\infty a_j) \overline{b_i}=\sum_{j=1}^\infty\sum_{i=1}^ja_j\overline{b_j}=\sum_{j=1}^\infty a_j\overline{\sum_{i=1}^j{b_j}}. $$ So $$ T^*(b)_j=\sum_{i=1}^j{b_j}. $$ But then $T^*(b)\not\in W$ for any nonzero $b$ (as it would have infinitely many nonzero entries). So $T^*$ does not exist.