Operator monotone functions, operator convex functions, and strongly operator convex functions

Lawrence G. Brown

9:30 am to 10:30 am, October 29th, 2014   Science Building A1510

Abstract:

The three terms denote classes of real-valued functions on intervals, each of which can be defined by matrix inequalities. All or almost all of the talk will be suitable for a general audience, including students, and the talk will include an explanation of what is meant by a matrix inequality. Operator monotone functions are defined by the inequality, $f(h_1) \le f(h_2)$, whenever $h_1$ and $h_2$ are self-adjoint matrices whose eigenvalues are in the domain of f and $h_1 \le h_2$ (the meaning of f(h) will also be explained); and the definitions of the other two classes are also very natural. Each of the three classes has other characterizations of four different types, a global condition on f, an integral representation of f, a differential criterion, and a characterization in terms of operator algebraic semicontinuity theory. The main thrust of the talk will be to explain the various characterizations in a parallel way.

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