Loewner Matrices

Let f be a smooth function on R. The divided difference matrices
whose (i,j) entries are [\\frac{f(\\lambda_i)-f(\\lambda_j)}{\\lambda_i-\\lambda_j}], \\|ambda_1,...,\\lambda_n \\in \\mathbb{R} are called Loewner matrices. In a seminal paper published in 1934 Loewner used properties of these matrices to characterise operator monotone functions. In the same paper he established connections between this matrix problem, complex analytic functions, and harmonic analysis. These elegant connections sent Loewner matrices into the background. Some recent work has brought them back into focus. In particular, characterisation of operator convex functions in terms of Loewner matrices has been obtained. In this talk we describe some of this work.