Representing operators in the Glauber-Sudarshan P-representation

Question

If $| \alpha >$ 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 >< \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><\alpha|\frac{d^2 \alpha}{\pi} \end{equation}

Both the integrals are over the entire complex plane.

Answer 1

The answer is no, and the details are clearly spelled out in Glauber's Les Houches lectures (circa 1964). Glauber introduces a "T-representation" which can represent any operator in the Fock space of harmonic oscillator states, a less general "R-representation" which can represent any density operator, and the still less general "P-representation" which can "represent virtually all [states] studied in optics".