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Moment inequalities in matrix algebras

Periodic Reporting for period 1 - Moments (Moment inequalities in matrix algebras)

Reporting period: 2015-09-01 to 2017-08-31

Moments have been intensively studied in probability theory, quantum information theory or matrix analysis. During the last years, non-commutative moment estimates have received considerable attention in the scientific community. Firstly, this phenomenon originates in studies on the extreme properties of the standard deviation in quantum information theory and its recent applications in commutator estimates. Secondly, there has been an urgent need driven by particle physics to develop a non-commutative version of the classical theory of compact metric spaces and the Gromov-Hausdorff distance. One building block on this road is the concept of strongly Leibniz seminorms which appears in the work of Rieffel on the problem of convergence of finitely generated modules over quantum metric spaces. Interestingly, the simplest example of such seminorms is the non-commutative standard deviation. On the other hand, we need to mention that one can find Leibniz-type inequalities in the general theory of Dirichlet forms and non-linear PDEs as well where they are known as the Kato-Ponce inequalities.

The primary goal of the research project is to provide a better understanding of non-commutative moment inequalities in terms of their commutative counterparts and the strongly Leibniz seminorms through new examples. Specifically, we addressed the following fundamental questions: How large the higher-order non-commutative moments can be? Do all central moments have the strong Leibniz property?

These questions are sitting at an exciting intersection of several mathematical disciplines, including linear algebra, singular value inequalities, matrix and functional analysis and rearrangement inequalities.

We completely solved the first question in case of the fourth central moment and proved that the corresponding commutative bound is, in fact, an upper bound in the general matrix case. However, examples show that the sharp upper bound can be strictly smaller for special matrices. Furthermore, we have provided several particular answers in the general case as well.

In regards to the second question, first, we studied the Leibniz inequality of central moments in ordinary probability spaces. We demonstrated that all central moments satisfy the Leibniz property, but they are not necessarily strongly Leibniz. We have been presenting several proofs to this problem that reveal a strong connection to rearrangement inequalities and open up the way to much more general theorems. Numerical simulations support the conjecture of the Leibniz property for singular values of Hermitian products as well. Our convexity approach to the problem has led to new proofs of several singular value inequalities in the literature.
Regular discussions with the staff of the faculty at Royal Holloway, visiting seminars, conferences and research schools have widened our knowledge in functional analysis, linear algebra and statistics. This activity indirectly led to novel ideas and solutions. Now we briefly summarise the achievements of the project.

In the article ‘Some inequalities for central moments of matrices’ we proved an estimate of the fourth moment for matrices. We obtained that the upper bound in the commutative case may serve as an upper bound in the non-commutative case as well. An example shows that the commutative bound is not necessarily sharp for all matrices, not like it is for the standard deviation. Several particular results have been obtained for general higher-order central moments.

As an objective of the project, we proved the Leibniz inequality in probability measure spaces and discussed the results in the frame of the Cipriani-Sauvageot differential calculus. In the spirit of the Kato-Ponce inequality, we have presented an unexpected extension of the previous results. Additionally, we have constructed an example which shows that the strong Leibniz property does not hold for all central moments. These results have been published under the gold open access policy to guarantee the free access to the broader public.

Interestingly, we have found a completely different look upon the Leibniz property through rearrangement inequalities and discussed the results in rearrangement invariant Banach function spaces as well. Investigating the non-commutative case, we developed a convexity method which turned out to be useful to prove certain singular values inequalities; however, the problem of Leibniz-type rules for Hermitian matrices has remained open.

Concerning the dissemination of the project, we have uploaded our articles to the worldwide available arXiv servers and Internet portals such as the ResearchGate or our webpage. Furthermore, we announced the results at several international conferences, workshops as well as university seminars. In order to introduce the topic to graduate and PhD students, we organised a series of lectures,‘Inequalities via majorizations,’ at Royal Holloway.
Our work has highlighted an unexpected connection between Leibniz-type inequalities of central moments and rearrangement inequalities. This link has already led to interesting results and may provide a new view on Leibniz-type rules; however, we think that the phenomenon has not been fully exploited. We pointed out that central moments can be computed through a first-order differential calculus. The results and the applied techniques of the proofs present intriguing connections between several issues of potential importance in non-commutative metric geometry. Our work led to some novel ideas in establishing singular value inequalities.

We believe that our methods and results will lead to theoretical impacts, and the societal implications of such a project are usually indirect or hidden and take time to be realized.