Convergence with respect to the graph norm

Question

Let $T:A\subset E\to F$ be a linear closed operator with dense domain. (E,F are Banach spaces) and $x_n$ a sequence in A such that $||x_n-x||_E\to 0$. Let $||x||_1:=||x||_E+||T(x)||_F$ be the graph norm in A (A is a Banach space with this norm). I want to find out if we can deduce that even $||x_n-x||_1\to 0$ holds. I would appreciate any idea. Thank you in advance.

Answer 1

No. For example, on $\ell^1$ let $(T x)(n) = n x(n)$, with $A = \{x \in \ell^1 \mid \sum_n n |x(n)| < \infty\}$. Take $x = 0$ and $x_n = e_n/n$, where $e_n$ is the sequence with $e_n(n) = 1$, $e_n(m) = 0$ otherwise.