Suppose I have a bounded linear operator $T: H_1 \to H_2$, where $H_1, H_2$ are Hilbert spaces. It then follows that $T^*:H_2 \to H_1$ is linear and bounded with $\lVert{T^*}\rVert = \lVert{T}\rVert$, where we define the adjoint by $$\langle T \phi, \psi \rangle_{H_2} = \langle \phi, T^* \psi \rangle_{H_1} $$
For $\phi \in H_1$ and $\psi \in H_2$. The specific situation I am considering is the following. The Laplacian operator is bounded from $H^2 \to L^2$, both of which are Hilbert spaces. Hence $\Delta^*: L^2 \to H^2$ is a bounded operator. What is $\Delta^*$? By elementary computations $\Delta^* = \Delta$ (since I expect the Laplacian to be symmetric) but this yields a non-sensical answer. Could anyone help clarify?
Edit: $\Delta^*$ has to satisfy: $$ \langle \Delta \phi, \psi \rangle = \langle \phi, \Delta^* \psi \rangle + \langle \nabla \phi, \nabla \Delta^* \psi \rangle + \langle \Delta \phi, \Delta \Delta^* \psi \rangle$$ for $\phi \in H^2$ and $\psi \in L^2$, where $\langle \cdot , \cdot \rangle$ denotes the $L^2$-inner product.