It is shown that for every pair of natural numbers $m\geq n\geq 1$, there exists a compact Fréchet space $X_{m,n}$ such that \item {(a)} $\mathop{\rm dim}X_{m,n}=n$, $\mathop{\rm ind}X_{m,n}=\mathop{\rm Ind}X_{m,n}=m$, and \item {(b)} every component of $X_{m,n}$ is homeomorphic to the $n$-dimensional cube $I^n$. \endgraf \noindent This yields new counter-examples to the theorem on dimension-lowering maps in the cases of inductive dimensions.
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 $[T_{\bar{z_i}},T_{z_i}]= [T^*_{{z_i}},T_{z_i}]\:\mathcal F_\mu \rightarrow L^2(\mu )$.
We study sub-Bergman Hilbert spaces in the weighted Bergman space $A^2_\alpha $. We generalize the results already obtained by Kehe Zhu for the standard Bergman space $A^2$.