Advanced Search

Commuting multi-contractions of class $C_{0·}$ and having a regular isometric dilation are studied. We prove that a polydisc contraction of class $C_{0·}$ is the restriction of a backwards multi-shift to an invariant subspace, extending a particular case of a result by R. E. Curto and F.-H. Vasilescu. A new condition on a commuting multi-operator, which is equivalent to the existence of a regular isometric dilation and improves a recent result of A. Olofsson, is obtained as a consequence.

Let $\varphi$ be a psh function on a bounded pseudoconvex open set $\Omega \subset {ℂ}^n$, and let $ℐ\left(m\varphi \right)$ be the associated multiplier ideal sheaves, $m\in {\mathbb{N}}^{☆}$. Motivated by global geometric issues, we establish an effective version of the coherence property of $ℐ\left(m\varphi \right)$ as $m\to +\infty$. Namely, given any $B⋐\Omega$, we estimate the asymptotic growth rate in $m$ of the number of generators of $ℐ{\left(m\varphi \right)}_{|B}$ over ${𝒪}_{\Omega }$, as well as the growth of the coefficients of sections in $\Gamma (B,ℐ(m\varphi \left)\right)$ with respect to finitely many generators globally defined on $\Omega$. Our approach relies on proving asymptotic integral...