Geodesic
The $\bar\partial$-Neumann operator and commutators of the Bergman projection and multiplication operators
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 4, pp. 1247-1256.

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We prove that compactness of the canonical solution operator to $\bar \partial $ restricted to $(0,1)$-forms with holomorphic coefficients is equivalent to compactness of the commutator $[\mathcal P,\bar M]$ defined on the whole $L^2_{(0,1)}(\Omega ),$ where $\bar M$ is the multiplication by $\bar z$ and $\mathcal P $ is the orthogonal projection of $L^2_{(0,1)}(\Omega )$ to the subspace of $(0,1)$ forms with holomorphic coefficients. Further we derive a formula for the $\bar \partial $-Neumann operator restricted to $(0,1)$ forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications operators by $z $ and $\bar z$.
Classification : 32A36, 32W05
Keywords: $\bar{\partial}$-equation; $\bar{\partial}$-Neumann operator; compactness 
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     author = {Haslinger, Friedrich},
     title = {The $\bar\partial${-Neumann} operator and commutators of the {Bergman} projection and multiplication operators},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1247--1256},
     publisher = {mathdoc},
     volume = {58},
     number = {4},
     year = {2008},
     mrnumber = {2471181},
     zbl = {1174.32015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2008__58_4_a28/}
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Haslinger, Friedrich. The $\bar\partial$-Neumann operator and commutators of the Bergman projection and multiplication operators. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 4, pp. 1247-1256. http://geodesic.mathdoc.fr/item/CMJ_2008__58_4_a28/