The von Neumann inequality says the value of a polynomial at a contractive operator is bounded by the norm of the polynomial on the disk. The von Neumann inequality is often proven using the Sz.-Nagy dilation theorem, which essentially says that one can model a contraction by a unitary operator. We adapt a technique of Nelson for proving the von Neumann inequality: one considers the singular value decomposition and then replaces the singular values with automorphisms of the disk to obtain a matrix valued analytic function which must attain its maximum on the boundary. Moreover, the matrix valued function involved in fact gives a minimal unitary dilation. With McCullough, we adapt Nelson’s trick to various other classes of operators to obtain their dilation theory, including the quantum annulus, row contractions and doubly commuting contractions. We conjecture a geometric relationship between Ando’s inequality and Gerstenhaber’s theorem.

The video of this talk is available on the IISc Math Department channel.

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Last updated: 25 Jan 2023