= Solution
The orbital <angular momentum operators> are
$$
\boxed{\hat L_i=\sum_{j,k}\varepsilon_{ijk}\hat x_j\hat p_k=-i\hbar\sum_{j,k}\varepsilon_{ijk}x_j\partial_k,\qquad \hat L^2=\sum_{i=1}^3\hat L_i^2.}
$$
In particular $\hat L_3=-i\hbar(x_1\partial_2-x_2\partial_1)$. For smooth compactly supported functions $\phi,\psi$, integration by parts produces no boundary term, and $\partial_2x_1=\partial_1x_2=0$. Hence
$$
\langle\phi,\hat L_3\psi\rangle=\langle\hat L_3\phi,\psi\rangle.
$$
This proves Hermiticity on the common test-function domain, and the same calculation applies on suitable decaying domains. As an unbounded operator, it is not defined on every $L^2$ function: the usual <self-adjoint operator> realization is the generator of unitary rotations about the third axis. Thus the integration statement includes its necessary domain/boundary assumptions.
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