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Register a new userStable division and essential normality: the non-homogeneous and quasi homogeneous cases
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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 $p>2$.
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In this paper, we show that under a mild condition, a principal submodule of the Bergman module on a bounded strongly pseudoconvex domain with smooth boundary in $mathbb{C}^n$ is $p$-essentially normal for all $p>n$. This is a significant improvement of the results of the first author and K. Wang, where the same result is shown to hold for polynomial-generated principal submodules of the Bergman module on the unit ball $mathbb{B}_n$ of $mathbb{C}^n$. As a consequence of our main result, we prove that the submodule of $L_a^2(mathbb{B}_n)$ consisting of functions vanishing on a pure analytic subsets of codimension $1$ is $p$-essentially normal for all $p>n$.
Let $S = (S_1, ldots, S_d)$ denote the compression of the $d$-shift to the complement of a homogeneous ideal $I$ of $mathbb{C}[z_1, ldots, z_d]$. Arveson conjectured that $S$ is essentially normal. In this paper, we establish new results supporting this conjecture, and connect the notion of essential normality to the theory of the C*-envelope and the noncommutative Choquet boundary. The unital norm closed algebra $mathcal{B}_I$ generated by $S_1,ldots,S_d$ modulo the compact operators is shown to be completely isometrically isomorphic to the uniform algebra generated by polynomials on $overline{V} := overline{mathcal{Z}(I) cap mathbb{B}_d}$, where $mathcal{Z}(I)$ is the variety corresponding to $I$. Consequently, the essential norm of an element in $mathcal{B}_I$ is equal to the sup norm of its Gelfand transform, and the C*-envelope of $mathcal{B}_I$ is identified as the algebra of continuous functions on $overline{V} cap partial mathbb{B}_d$, which means it is a complete invariant of the topology of the variety determined by $I$ in the ball. Motivated by this determination of the C*-envelope of $mathcal{B}_I$, we suggest a new, more qualitative approach to the problem of essential normality. We prove the tuple $S$ is essentially normal if and only if it is hyperrigid as the generating set of a C*-algebra, which is a property closely connected to Arvesons notion of a boundary representation. We show that most of our results hold in a much more general setting. In particular, for most of our results, the ideal $I$ can be replaced by an arbitrary (not necessarily homogeneous) invariant subspace of the $d$-shift.
We study density requirements on a given Banach space that guarantee the existence of subsymmetric basic sequences by extending Tsirelsons well-known space to larger index sets. We prove that for every cardinal $kappa$ smaller than the first Mahlo cardinal there is a reflexive Banach space of density $kappa$ without subsymmetric basic sequences. As for Tsirelsons space, our construction is based on the existence of a rich collection of homogeneous families on large index sets for which one can estimate the complexity on any given infinite set. This is used to describe detailedly the asymptotic structure of the spaces. The collections of families are of independent interest and their existence is proved inductively. The fundamental stepping up argument is the analysis of such collections of families on trees.
With an eye toward understanding complexity control in deep learning, we study how infinitesimal regularization or gradient descent optimization lead to margin maximizing solutions in both homogeneous and non-homogeneous models, extending previous work that focused on infinitesimal regularization only in homogeneous models. To this end we study the limit of loss minimization with a diverging norm constraint (the constrained path), relate it to the limit of a margin path and characterize the resulting solution. For non-homogeneous ensemble models, which output is a sum of homogeneous sub-models, we show that this solution discards the shallowest sub-models if they are unnecessary. For homogeneous models, we show convergence to a lexicographic max-margin solution, and provide conditions under which max-margin solutions are also attained as the limit of unconstrained gradient descent.
In our paper Essential normality, essential norms and hyperrigidity we claimed that the restriction of the identity representation of a certain operator system (constructed from a polynomial ideal) has the unique extension property, however the justification we gave was insufficient. In this note we provide the required justification under some additional assumptions. Fortunately, homogeneous ideals that are sufficiently non-trivial are covered by these assumptions. This affects the section of our paper relating essential normality and hyperrigidity. We show here that Proposition 4.11 and Theorem 4.12 hold under the additional assumptions. We do not know if they hold in the generality considered in our paper.
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