Colloquium

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 graduate 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.