Description
CCOR Optimalizálási Szemináriumo
Abstract:
In the context of free function theory, we review recent results on operator monotone and concave functions in several variables. We establish representation formulas using LMIs and the theory of matrix convex sets. For an operator concave function we define its hypograph as the downward saturation of its graph with respect to the positive definite order. Then operator concavity of a free function is characterized by the matrix convexity of its hypograph. Given a closed matrix convex hypograph as a subset of a linear space of bounded linear operators, one can find its supporting linear functionals and represent them as linear pencils of operators on the tensor product of the linear space with its dual space. Then the linear pencil defines a semidefinite program, such that its extremal solution coincides with the operator concave function. We exploit this to construct a non-commutative analytic lift for partially defined functions such as multivariable real functions that are either operator monotone or operator concave. These results can also be applied in more general settings, like in the case of operator means of probability measures.
Aki szeretne csatlakozni, kérem, hogy e-mailben jelezze Varga Anitánál (vanita@math.bme.hu).