Intertwining Liftings and Young Tableaux Related Problems

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Abstract: This project is to try to obtain a better understanding of the structure of bounded linear operators acting on a complex, separable, infinite dimensional Hilbert space. The principal investigator plans to continue her research on the theory of commutant lifting theorem, including the numerical aspect and the generalization to the multioperator setting. At the same time, the principal investigator will continue her research on sums of eigenvalues of self-adjoint operators, and its connections with intersection theory in algebraic geometry, Young tableaux, and representation theory. This project is to try to obtain a better understanding of the theory of linearoperators acting on Hilbert space (``operator theory''). There are two main research areas that this proposal will focus on. One is in the area of the theory of commutant lifting thorem, which has proven to be useful in control theory in engineering. The other one is on a problem about eigenvalues of sums of (self-adjoint) operators. This problem originated in a celebrated paper of Weyl in 1912 while he was studying partial differential equations. Only recently that there was a significant break through and the solution depends on some very deep theorems in algebraic geometry. It is our hope that this project will bring some insight into the problem from the operator theory point of view. Furthermore we want to extend the existing theory to a setting that can be applied to quantum physics.