BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Realization theory for Herglotz--Nevanlinna and Schur class functions: from one to several variables and beyond Vinnikov, Victor


In this survey talk I will recall how the familiar integral representation for Herglotz--Nevanlinna functions can be recast in operator theoretic or rather system theoretic terms, namely as a realization of the function as the transfer function of an input/state/output linear system. There are similar results for Schur functions, i.e., functions taking values in the unit disc (as opposed to the upper or the right halfplane). After discussing the one variable case, I will move to the case of several complex variables. It turns out that there is a very good generalization of the classical realization theory provided one considers possibly more restricted classes of functions. These classes are defined by testing the values of the function not merely on tuples of scalars but on tuples of commuting matrices of all sizes. Time permitting I will wrap up with the most recent incarnation of these ideas, where one tests the values on tuples of not necessarily commuting matrices.

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