Searcharxiv⌕ Search

arXiv subjects

Zoltan Leka

Publications and source records attributed to Zoltan Leka.

7 recordsLinked to original sources
arXiv2017

Some singular value inequalities via convexity

If $c_1(Z) \geq ... \geq c_n(Z)$ denote the Euclidean lengths of the column vectors of any $n \times n$ matrix $Z,$ then a fundamental inequality related to Hadamard products states that $$ \sum_{i=1}^k σ_i(X^*Y \circ B) \leq \sum_{i=1}^k c_i(X) c_i(Y) σ_i(B) \qquad 1 \leq k \leq n,$$ where $σ_i(\cdot)$ is the $i$th singular value. In this paper, we shall offer a simple proof of this result via convexity arguments. In addition, this technique is applied to obtain some further singular value inequalities as well.

math.FA↗
arXiv2017

On the Leibniz rule for random variables

We prove a Leibniz-type inequality for the spread of random variables in terms of their $L_p$-norms. The result is motivated by the Kato-Ponce inequalities and Rieffel's strong Leibniz property.

math.FA↗
arXiv2016

Symmetric norms and the Leibniz property

We show that certain symmetric seminorms on $\mathbb{R}^n$ satisfy the Leibniz inequality. As an application, we obtain that $L^p$ norms of centered bounded real functions, defined on probability spaces, have the same property. Even though this is well-known for the standard deviation it seems that the complete result has never been established. In addition, we shall connect the results with the differential calculus introduced by Cipriani and Sauvageot and Rieffel's non-commutative Riemann metric.

math.FA↗
arXiv2015

Some inequalities for central moments of matrices

In this paper we shall study noncommutative central moment inequalities with a main focus on whether the commutative bounds are tight in the noncommutative case, or not. We prove that the answer is affirmative for the fourth central moment and several particular results are given in the general case. As an application, we shall present some lower estimates of the spread of Hermitian and normal matrices as well.

math.FA↗
arXiv2014

Leibniz seminorms in probability spaces

In this paper we study the (strong) Leibniz property of centered moments of bounded random variables. We shall answer a question raised by M. Rieffel on the non-commutative standard deviation.

math.FA↗
arXiv2014

A note on extremal decomposition of covariances

We shall present an elementary approach to extremal decompositions of (quantum) covariance matrices determined by densities. We give a new proof on former results and provide a sharp estimate of the ranks of the densities that appear in the decomposition theorem.

math.FA↗