I have some confusion about the definition of self-adjoint operators and formally self-adjoint operators. Let me write down the background information.
Let H be a infinite dimensional complex Hilbert space and T:D(T)\to H a (not necessarily bounded) linear operator , where D(T) is dense in H. The operator T is said to be formally self-adjoint if for all x, y\in D(T), we have
\langle Tx, y\rangle=\langle x, Ty\rangle.
The operator T is said to be self-adjoint if T^*=T, where T^* is the adjoint of T. Of course if D(T)=H and T is formally self-adjoint in the above sense, then T must be bounded and therefore T is also self adjoint.
Let M be a closed Riemannian manifold and P:C^\infty(M)\to C^\infty(M) an elliptic pseudodifferential operator. We know C^\infty(M) is dense in L^2(M) (and even in H_s(M)).
My confusion is the following. Suppose P is formally self-adjoint, i.e., for all f, g\in C^\infty(M), we have
\langle Pf, g\rangle_{L^2(M)}=\langle f, Pg\rangle_{L^2(M)}.
We also know that the extension of P to P:H_s(M)\to H_{s-d}(M) is a bounded linear operator (indeed it's Fredholm). Well, then I don't know how to continue my question, perhaps I am really confused by my confusion. Anyway, I would appreciate if someone can clear my confusion for which I couldn't even explain.
Thanks~
