We study the spectral properties of Jacobi matrices. By combining Killip's technique [12] with the technique of Killip and Simon [13] we obtain a result relating the properties of the elements of Jacobi matrices and the corresponding spectral measures. This theorem is a natural extension of a recent result of Laptev-Naboko-Safronov [17].

Let $\mathcal{D}$ be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We give a new sufficient condition, not far from the known necessary condition, for a function$f$∈ $\mathcal{D}$ to be$cyclic$, i.e. for {$pf$:$p$is a polynomial} to be dense in $\mathcal{D}$ .

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Let ($X, L$) be a polarized 3-fold over the complex number field. In [Fk3], we proved that$g(L)≥q(X)$if$h$^{0}($L$)≥2 and moreover we classified ($X, L$) with$h$^{0}($L$)≥3 and$g(L)$=$q(X)$, where$g(L)$is the sectional genus of ($X, L$) and$q$($X$)=dim$H$^{1}($O$_{$X$}) the irregularity of$X$. In this paper we will classify polarized 3-folds ($X, L$) with$h$^{0}($L$)≥4 and$g(L)$=$q(X)$+1 by the method of [Fk3].

Let$X$be a complex manifold and let$f$:$X$→ℂ^{$p$}be a holomorphic mapping defining a complete intersection. We prove that the iterated Mellin transform of the residue integral associated with$f$has an analytic continuation to a neighborhood of the origin in ℂ^{$p$}.