Thursday, February 8 2024
14:00 - 15:00

Chandrasekhar Hall

Spectral decompositions of operators in tracial von Neumann algebras

Amudhan Krishnaswamy Usha

l;a;allNational Institute of Standards and Technology, USA

As a consequence of the Jordan canonical form, every matrix A is similar to a matrix of the form N+Q, where N is diagonal, Q is nilpotent (has spectrum ={0}) and N,Q commute with each other. If one is interested in operators on infinite dimensional Hilbert spaces which behave "like matrices", it is natural to look for similar decompositions. Here we ask that N is normal, Q has spectrum {0} (i.e. Q is quasi-nilpotent), and N, Q commute. Dunford characterized such operators in terms of the existence of certain subspaces, and called them `spectral operators'.

In many respects, von Neumann algebras which possess a trace behave like matrix algebras. For instance, operators living in such algebras have analogues of generalized eigenspaces, called the Haagerup-Schultz subspaces. I will show how one can characterize spectral operators in this setting in terms of the angles between these subspaces. Moreover, I will show how these angles are actually computable for many operators arising from random matrix and free probability theory, such as Voiculescu's circular operator. 

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