Abstract. In their paper ''Positivity conditions for bihomogeneous polynomials", D. Catlin and J. P. D'Angelo use compactness of the commutators $[M_{\varphi},P]$ in conjunction with their study of a complex variables analogue of Hilbert's 17th problem.
In particular, they show that the compactness of $N_1$ implies that the commutators $[M,P]$ are compact for all tangential pseudodifferential operators $M$ of order $0$. I will present this result, along with some properties of the $\bar \partial$-Neumann operator.