We prove Obata’s rigidity theorem for metric measure spaces that satisfy a Riemannian curvaturedimension condition. Additionally,we show that a lower bound K for the generalizedHessian of a sufficiently regular function u holds if and only if u is K-convex. A corollary is also a rigidity result for higher order eigenvalues.
In this paper, we prove a result linking the square and the rectangular R-transforms, the consequence of which is a surprising relation between the square and rectangular versions the free additive convolutions, involving the Marchenko–Pastur law. Consequences on random matrices, on infinite divisibility and on the arithmetics of the square versions of the free additive and multiplicative convolutions are given.
We show that in the space C[-1,1] there exists an orthogonal algebraic polynomial basis with optimal growth of degrees of the polynomials.
The paper is, for the most part, devoted to a survey of the analytical properties of generalized convolution algebras and their realizations. This issue appears to be the state of the art until now because intensive research on the generalized convolution and the related models still persists.
Consider a non-centered matrix with a separable variance profile: Matrices and are non-negative deterministic diagonal, while matrix is deterministic, and is a random matrix with complex independent and identically distributed random variables, each with mean zero and variance one. Denote by the resolvent associated to , i.e. Given two sequences of deterministic vectors and with bounded Euclidean norms, we study the limiting behavior of the random bilinear form: as the dimensions...