If $| \alpha\rangle $ represents a coherent state (the normalized right eigenstate of the destruction operator $a$ in Quantum Mechanics; $\alpha$ is a complex number), then it is known that:
\begin{equation} \int | \alpha\rangle\!\langle \alpha| \frac{d^2\alpha}{\pi} = I \end{equation} where $I$ refers to the identity operator.
Can any operator acting on the appropriate Hilbert space be represented in the Glauber-Sudarshan P-representation, and if it can, how to prove that this is the case? (I am especially interested about the representation of density operators)
By the Glauber-Sudarshan representation, I mean the following: \begin{equation} \int P(\alpha,\alpha^*)|\alpha\rangle\!\langle\alpha|\frac{d^2 \alpha}{\pi} \end{equation}
Both the integrals are over the entire complex plane.
