The quasi-canonical solution operator to $\overline{\partial }$ restricted to the Fock-space

@article{Schneider2005,
abstract = {We consider the solution operator $S\:\mathcal \{F\}_\{\mu ,(p,q)\}\rightarrow L^2(\mu )_\{(p,q)\}$ to the $\bar\{\partial \}$-operator restricted to forms with coefficients in $\mathcal \{F\}_\{\mu \}= \bigl \lbrace f\: f \text\{is\} \text\{entire\} \text\{and\} \int _\{\mathbb \{C\}^n\} |f(z)|^2\mathrm \{d\}\mu (z) <\infty \bigr \rbrace $. Here $\mathcal \{F\}_\{\mu ,(p,q)\}$ denotes $(p,q)$-forms with coefficients in $\mathcal \{F\}_\{\mu \}$, $L^2(\mu )$ is the corresponding $L^2$-space and $\mu $ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula $S$ to $\bar\{\partial \}$. This solution operator will have the property $Sv\bot \mathcal \{F\}_\{(p,q)\}\, \forall \,v \in \mathcal \{F\}_\{(p,q+1)\}$. As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness of commutators of Toeplitz-operators $[T_\{\bar\{z_i\}\},T_\{z_i\}]= [T^*_\{\{z_i\}\},T_\{z_i\}]\:\mathcal \{F\}_\mu \rightarrow L^2(\mu )$.},
author = {Schneider, Georg},
journal = {Czechoslovak Mathematical Journal},
keywords = {Fock-space; Hankel-operator; reproducing kernel; Fock space; Hankel operator; reproducing kernel},
language = {eng},
number = {4},
pages = {947-956},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The quasi-canonical solution operator to $\bar\{\partial \}$ restricted to the Fock-space},
url = {http://eudml.org/doc/31002},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Schneider, Georg
TI - The quasi-canonical solution operator to $\bar{\partial }$ restricted to the Fock-space
JO - Czechoslovak Mathematical Journal
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 4
SP - 947
EP - 956
AB - We consider the solution operator $S\:\mathcal {F}_{\mu ,(p,q)}\rightarrow L^2(\mu )_{(p,q)}$ to the $\bar{\partial }$-operator restricted to forms with coefficients in $\mathcal {F}_{\mu }= \bigl \lbrace f\: f \text{is} \text{entire} \text{and} \int _{\mathbb {C}^n} |f(z)|^2\mathrm {d}\mu (z) <\infty \bigr \rbrace $. Here $\mathcal {F}_{\mu ,(p,q)}$ denotes $(p,q)$-forms with coefficients in $\mathcal {F}_{\mu }$, $L^2(\mu )$ is the corresponding $L^2$-space and $\mu $ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula $S$ to $\bar{\partial }$. This solution operator will have the property $Sv\bot \mathcal {F}_{(p,q)}\, \forall \,v \in \mathcal {F}_{(p,q+1)}$. As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness of commutators of Toeplitz-operators $[T_{\bar{z_i}},T_{z_i}]= [T^*_{{z_i}},T_{z_i}]\:\mathcal {F}_\mu \rightarrow L^2(\mu )$.
LA - eng
KW - Fock-space; Hankel-operator; reproducing kernel; Fock space; Hankel operator; reproducing kernel
UR - http://eudml.org/doc/31002
ER -