Example of an unbounded operator whose adjoint is not densely defined

Question

In his book "Quantum Theory for Mathematicians", B. C. Hall mentions that there are some pathological examples of unbounded operators on separable Hilbert spaces whose adjoint is not densely defined (p. 171). What is such an example?

Answer 1

The adjoint $A^*$ of an unbounded operator $A$ on a Hilbert space $\mathcal H$ is densely defined if and only if $A$ is closable, i.e. the closure of the graph $\Gamma(A) = \{(x, Ax) \in \mathcal H^2: x \in \mathcal D(A)\}$ is the graph of an operator. So for a counterexample all we need is a sequence $x_n \in \mathcal D(A)$ such that $x_n \to 0$ while $A x_n$ converges to some $y \in \mathcal H$. For example, if $e_n$ is an orthonormal sequence in $\mathcal H$ you could take $A e_n = n e_1$.