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Register a new userOn the symmetrized arithmetic-geometric mean inequality for opertors
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We study the symmetrized noncommutative arithmetic geometric mean inequality introduced(AGM) by Recht and R{e} $$ |frac{(n-d)!}{n!}sumlimits_{{ j_1,...,j_d mbox{ different}} }A_{j_{1}}^*A_{j_{2}}^*...A_{j_{d}}^*A_{j_{d}}...A_{j_{2}}A_{j_{1}} | leq C(d,n) |frac{1}{n} sum_{j=1}^n A_j^*A_j|^d .$$ Complementing the results from Recht and R{e}, we find upper bounds for C(d,n) under additional assumptions. Moreover, using free probability, we show that $C(d, n) > 1$, thereby disproving the most optimistic conjecture from Recht and R{e}.We also prove a deviation result for the symmetrized-AGM inequality which shows that the symmetric inequality almost holds for many classes of random matrices. Finally we apply our results to the incremental gradient method(IGM).
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In a recent work, Moslehian and Rajic have shown that the Gruss inequality holds for unital n-positive linear maps $phi:mathcal A rightarrow B(H)$, where $mathcal A$ is a unital C*-algebra and H is a Hilbert space, if $n ge 3$. They also demonstrate that the inequality fails to hold, in general, if $n = 1$ and question whether the inequality holds if $n=2$. In this article, we provide an affirmative answer to this question.
We present several operat
Let $E$ be a Bedford-McMullen carpet associated with a set of affine mappings ${f_{ij}}_{(i,j)in G}$ and let $mu$ be the self-affine measure associated with ${f_{ij}}_{(i,j)in G}$ and a probability vector $(p_{ij})_{(i,j)in G}$. We study the asymptotics of the geometric mean error in the quantization for $mu$. Let $s_0$ be the Hausdorff dimension for $mu$. Assuming a separation condition for ${f_{ij}}_{(i,j)in G}$, we prove that the $n$th geometric error for $mu$ is of the same order as $n^{-1/s_0}$.
We consider a version of the fractional Sobolev inequality in domains and study whether the best constant in this inequality is attained. For the half-space and a large class of bounded domains we show that a minimizer exists, which is in contrast to the classical Sobolev inequalities in domains.
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