id: 05876326
dt: j
an: 05876326
au: Shalit, Orr Moshe
ti: Stable polynomial division and essential normality of graded Hilbert
    modules.
so: J. Lond. Math. Soc., II. Ser. 83, No. 2, 273-289 (2011).
py: 2011
pu: London Mathematical Society, London; Oxford University Press, Oxford
la: EN
cc: 
ut: Arveson’s resistant conjecture; graded Hilbert module; d-shift Hilbert
    module; essentially normal; stable division property; ring of
    polynomials
ci: Zbl 1107.47006; Zbl 1108.47030; Zbl 1148.47005; Zbl 1140.47060
li: doi:10.1112/jlms/jdq054
ab: {\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)].
rv: Maryam Amyari (Mashhad)