In this note we establish some general finiteness results concerning lattices Γ in connected Lie groups$G$which possess certain “density” properties (see$Moskowitz$, M., On the density theorems of Borel and Furstenberg,$Ark. Mat.$$16$(1978), 11–27, and$Moskowitz$, M., Some results on automorphisms of bounded displacement and bounded cocycles,$Monatsh. Math.$$85$(1978), 323–336). For such groups we show that Γ always has finite index in its normalizer$N$_{$G$}(Γ). We then investigate analogous questions for the automorphism group Aut($G$) proving, under appropriate conditions, that Stab_{Aut($G$)}(Γ) is discrete. Finally we show, under appropriate conditions, that the subgroup $\tilde{\Gamma}=\{i_{\gamma}:\gamma \in \Gamma \},\ i_{\gamma}(x)=\gamma x\gamma^{-1}$ , of Aut($G$) has finite index in Stab_{Aut($G$)}(Γ). We test the limits of our results with various examples and counterexamples.

In this note, we characterize maximal invariant subspaces for a class of operators. Let$T$be a Fredholm operator and $1-TT^{*}\in\mathcal{S}_{p}$ for some$p$≥1. It is shown that if$M$is an invariant subspace for$T$such that dim$M$$⊖$$TM$<∞, then every maximal invariant subspace of$M$is of codimension 1 in$M$. As an immediate consequence, we obtain that if$M$is a shift invariant subspace of the Bergman space and dim$M$$⊖$$zM$<∞, then every maximal invariant subspace of$M$is of codimension 1 in$M$. We also apply the result to translation operators and their invariant subspaces.

We consider second order elliptic divergence form systems with complex measurable coefficients$A$that are independent of the transversal coordinate, and prove that the set of$A$for which the boundary value problem with$L$_{2}Dirichlet or Neumann data is well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when$A$is either Hermitean, block or constant. Our methods apply to more general systems of partial differential equations and as an example we prove perturbation results for boundary value problems for differential forms.

In [$Rong$, F., Quasi-parabolic analytic transformations of$C$^{$n$},$J. Math. Anal. Appl.$$343$(2008), 99–109], we showed the existence of “parabolic curves” for certain quasi-parabolic analytic transformations of$C$^{$n$}. Under some extra assumptions, we show the existence of “parabolic manifolds” for such transformations.

Let$α$_{1},$α$_{2},…,$α$_{$m$}be linear forms defined on ℂ^{$n$}and $\mathcal{X}=\mathbb{C}^{n}\setminus\bigcup_{i=1}^{m}V(\alpha_{i})$ , where$V$($α$_{$i$})={$p$∈ℂ^{$n$}$:$$α$_{$i$}($p$)=0}. The coordinate ring $O_{\mathcal{X}}$ of $\mathcal{X}$ is a holonomic$A$_{$n$}-module, where$A$_{$n$}is the$n$th Weyl algebra and since holonomic$A$_{$n$}-modules have finite length, $O_{\mathcal{X}}$ has finite length. We consider a “twisted” variant of this$A$_{$n$}-module which is also holonomic. Define M_{$α$}^{$β$}to be the free rank-1 ℂ[$x$]_{$α$}-module on the generator$α$^{$β$}(thought of as a multivalued function), where $\alpha^{\beta}=\alpha_{1}^{\beta_{1}},\ldots,\alpha_{m}^{\beta_{m}}$ and the multi-index$β$=($β$_{1},…,$β$_{$m$})∈ℂ^{$m$}. Our main result is the computation of the number of decomposition factors of M_{$α$}^{$β$}and their description when$n$=2.