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\iteman{io-port 05876326}
\itemau{Shalit, Orr Moshe}
\itemti{Stable polynomial division and essential normality of graded Hilbert modules.}
\itemso{J. Lond. Math. Soc., II. Ser. 83, No. 2, 273-289 (2011).}
\itemab
{\it W. Arveson} [J. Oper. Theory 54, No. 1, 101--117 (2005; Zbl 1107.47006)] conjectured that every graded submodule $M$ of $H_d^2\otimes \mathbb{C}^r$ and its quotient $H_d^2\otimes\mathbb{C}^r/M$ are $p$-essentially normal for $p > d$. This conjecture has been verified for modules generated by monomials [see {\it R. G. Douglas}, J. Oper. Theory 55, No. 1, 117--133 (2006; Zbl 1108.47030)] and also for principal modules as well as arbitrary modules in dimensions $d = 2, 3$ [cf. {\it K.-Y. Guo} and {\it K. Wang}, Math. Ann. 340, No. 4, 907--934 (2008; Zbl 1148.47005)]. In order to tackle this conjecture, the author introduces the stable division property for modules (and ideals); a normed module $M$ over the ring of polynomials in $d$ variables is said to possess the stable division property if it has a generating set $\{f_1, \dots, f_k\}$ such that every $h \in M$ can be written as $h =\sum_ia_if_i$ for some polynomials $a_i$ such that $\sum\|a_if_i\| \leq C\|h\|$. He shows that when the algebra of polynomials in $d$ variables is given the natural $\ell^1$-norm, then every ideal is linearly equivalent to an ideal that has the stable division property. He then proves that a module $M$ that has the stable division property is $p$-essentially normal for $p > \dim(M)$, as conjectured by {\it R. G. Douglas} [``A new kind of index theorem", in: Analysis, geometry and topology of elliptic operators. Papers of a workshop in honor of Krzysztof P.\ Wojciechowski on his 50th birthday, Roskilde, Denmark, May 20--22, 2005 (Hackensack, NJ: World Scientific), 369--382 (2006; Zbl 1140.47060)].
\itemrv{Maryam Amyari (Mashhad)}
\itemcc{}
\itemut{Arveson's resistant conjecture; graded Hilbert module; d-shift Hilbert module; essentially normal; stable division property; ring of polynomials}
\itemli{doi:10.1112/jlms/jdq054}
\end