Stable division and essential normality: the non-homogeneous and quasi homogeneous cases
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| Format: | Preprint |
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2015
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| author | Biswas, Shibananda Shalit, Orr |
| author_facet | Biswas, Shibananda Shalit, Orr |
| contents | Let $\mathcal{H}_d^{(t)}$ ($t \geq -d$, $t>-3$) be the reproducing kernel Hilbert space on the unit ball $\mathbb{B}_d$ with kernel \[ k(z,w) = \frac{1}{(1-\langle z, w \rangle)^{d+t+1}} . \] We prove that if an ideal $I \triangleleft \mathbb{C}[z_1, \ldots, z_d]$ (not necessarily homogeneous) has what we call the "approximate stable division property", then the closure of $I$ in $\mathcal{H}_d^{(t)}$ is $p$-essentially normal for all $p>d$. We then show that all quasi homogeneous ideals in two variables have the stable division property, and combine these two results to obtain a new proof of the fact that the closure of any quasi homogeneous ideal in $\mathbb{C}[x,y]$ is $p$-essentially normal for $p>2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1504_03465 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Stable division and essential normality: the non-homogeneous and quasi homogeneous cases Biswas, Shibananda Shalit, Orr Functional Analysis Operator Algebras 47A13, 47B32 (Primary), 12Y05, 13P10 (Secondary) Let $\mathcal{H}_d^{(t)}$ ($t \geq -d$, $t>-3$) be the reproducing kernel Hilbert space on the unit ball $\mathbb{B}_d$ with kernel \[ k(z,w) = \frac{1}{(1-\langle z, w \rangle)^{d+t+1}} . \] We prove that if an ideal $I \triangleleft \mathbb{C}[z_1, \ldots, z_d]$ (not necessarily homogeneous) has what we call the "approximate stable division property", then the closure of $I$ in $\mathcal{H}_d^{(t)}$ is $p$-essentially normal for all $p>d$. We then show that all quasi homogeneous ideals in two variables have the stable division property, and combine these two results to obtain a new proof of the fact that the closure of any quasi homogeneous ideal in $\mathbb{C}[x,y]$ is $p$-essentially normal for $p>2$. |
| title | Stable division and essential normality: the non-homogeneous and quasi homogeneous cases |
| topic | Functional Analysis Operator Algebras 47A13, 47B32 (Primary), 12Y05, 13P10 (Secondary) |
| url | https://arxiv.org/abs/1504.03465 |