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Register a new userDistributional inequalities for noncommutative martingales
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We establish distributional estimates for noncommutative martingales, in the sense of decreasing rearrangements of the spectra of unbounded operators, which generalises the study of distributions of random variables. Our results include distribution
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This paper builds on the theory of generalised functions begun in [1]. The Colombeau theory of generalised scalar fields on manifolds is extended to a nonlinear theory of generalised tensor fields which is diffeomorphism invariant and has the sheaf property. The generalised Lie derivative for generalised tensor fields is introduced and it is shown that this commutes with the embedding of distributional tensor fields. It is also shown that the covariant derivative of generalised tensor fields commutes with the embedding at the level of association. The concept of generalised metric is introduced and used to develop a nonsmooth theory of differential geometry. It is shown that the embedding of a continuous metric results in a generalised metric with well defined connection and curvature. It is also shown that a twice continuously differentiable metric which is a solution of the vacuum Einstein equations may be embedded into the algebra of generalised tensor fields and has generalised Ricci curvature associated to zero. Thus, the embedding preserves the Einstein equations at the level of association. Finally, we consider an example of a metric which lies outside the Geroch-Traschen class and show that in our diffeomorphism invariant theory the curvature of a cone is associated to a delta function.
Let $T=begin{bmatrix} X &Y 0 & Zend{bmatrix}$ be an $n$-square matrix, where $X, Z$ are $r$-square and $(n-r)$-square, respectively. Among other determinantal inequalities, it is proved $det(I_n+T^*T)ge det(I_r+X^*X)cdot det(I_{n-r}+Z^*Z)$ with equality holds if and only if $Y=0$.
We show norm estimates for the sum of independent random variables in noncommutative $L_p$-spaces for $1<p<infty$ following our previous work. These estimates generalize the classical Rosenthal inequality in the commutative case. Among applications, we derive an equivalence for the $p$-norm of the singular values of a random matrix with independent entries, and characterize those symmetric subspaces and unitary ideals which can be realized as subspaces of a noncommutative $L_p$ for $2<p<infty$.
Following the recent work of Jiang and Lin (Linear Algebra Appl. 585 (2020) 45--49), we present more results (bounds) on Harnack type inequalities for matrices in terms of majorization (i.e., in partial products) of eigenvalues and singular values. We discuss and compare the bounds derived through different ways. Jiang and Lins results imply Tungs version of Harnacks inequality (Proc. Amer. Math. Soc. 15 (1964) 375--381); our results %with simpler proofs are stronger and more general than Jiang and Lins. We also show some majorization inequalities concerning Cayley transforms. Some open problems on spectral norm and eigenvalues are proposed.
Let $gamma$ be the standard Gaussian measure on $mathbb{R}^n$ and let $mathcal{P}_{gamma}$ be the space of probability measures that are absolutely continuous with respect to $gamma$. We study lower bounds for the functional $mathcal{F}_{gamma}(mu) = {rm Ent}(mu) - frac{1}{2} W^2_2(mu, u)$, where $mu in mathcal{P}_{gamma}, u in mathcal{P}_{gamma}$, ${rm Ent}(mu) = int logbigl( frac{mu}{gamma}bigr) d mu$ is the relative Gaussian entropy, and $W_2$ is the quadratic Kantorovich distance. The minimizers of $mathcal{F}_{gamma}$ are solutions to a dimension-free Gaussian analog of the (real) Kahler-Einstein equation. We show that $mathcal{F}_{gamma}(mu) $ is bounded from below under the assumption that the Gaussian Fisher information of $ u$ is finite and prove a priori estimates for the minimizers. Our approach relies on certain stability estimates for the Gaussian log-Sobolev and Talagrand transportation inequalities.
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