We describe spectral and decomposable operators on Hilbert spaces and then, in the case of operators belonging to finite von Neumann algebras, relate these properties to the Haagerup-Schultz subspaces of the operators. Finally, we will examine certain such operators arising naturally in free probability theory.

We describe spectral and decomposable operators on Hilbert spaces and then, in the case of operators belonging to finite von Neumann algebras, relate these properties to the Haagerup-Schultz subspaces of the operators. Finally, we will examine certain such operators arising naturally in free probability theory.

We give a non-asymptotic estimate for the spectral norm of a large class of random matrices that is sharp in many cases. We also obtain strong asymptotic freeness for certain sparse Gaussian matrices. Joint work with Afonso Bandeira and Ramon van Handel.

References:

• https://arxiv.org/abs/1504.05919
• https://arxiv.org/abs/2108.0631
• https://arxiv.org/abs/2208.11286

ZOOM LINK: https://tamu.zoom.us/j/97389162643

I will begin by proving an estimate for a quantity introduced by Tropp. I will then give a combinatorial proof and an analytic proof of the main result in my previous talk.

References:

• https://arxiv.org/abs/2104.02662
• https://arxiv.org/abs/2108.06312

ZOOM LINK: https://tamu.zoom.us/j/97389162643

It is well-known that many random matrices have an asymptotical limit which is described by free probability. That is, for any noncommutative polynomial in these d independent random matrices converges to the same polynomial in d freely independent random variables that describe the limit distribution of each sequence of random matrices. In this talk, we will present a natural generalization of this convergence result. Namely, under suitable assumptions, we can enlarge our test function from noncommutative polynomial to noncommutative rational functions. It is based on a joint-work with Benoît Collins, Tobias Mai, Akihiro Miyagawa and Félix Parraud.