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Sarason's one variable commutant lifting theorem is a key result in the theory of linear operators, complex analysis, and Hilbert function space theory, which has a stellar reputation in its application to classical results like Nevanlinna-Pick interpolation, Caratheodory-Fejer interpolation problem, Nehari interpolation problem, von Neumann inequality, isometric dilations, just to name a few. The expanded list easily includes control theory and electrical engineering. However, Sarason's lifting theorem does not hold in the setting of polydisc in general. Comprehending the subtleties of the lifting theorem on the polydisc is considered to be one of the challenging problems.
In the first half of this talk, we will provide a quick historical overview (within the span of little more than a century), present an introduction to the commutant lifting theorem, and explore how it interacts with the Nevanlinna-Pick interpolation. The second half of the talk will go over some recent advances in the commutant lifting theorem on the polydisc and its applications to interpolation and perturbation problems. |