Regular matrices of unbounded linear operators

Let $X,Y$ be Banach spaces, and fix a linear operator $T \in \mathcal{L}(X,Y)$, and ideals $\mathcal{I}, \mathcal{J}$ on $\omega$. We obtain Silverman--Toeplitz type theorems on matrices $A=(A_{n,k}: n,k \in \omega)$ of linear operators in $\mathcal{L}(X,Y)$, so that $$\mathcal{J}\text{-}\lim Ax=T(\hspace{.2mm}\mathcal{I}\text{-}\lim x)$$ for every $X$-valued sequence $x=(x_0,x_1,\ldots)$ which is $\mathcal{I}$-convergent [and bounded]. This allows us to establish the relationship between the classical Silverman--Toeplitz characterization of regular matrices and its multidimensional analogue for double sequences, its variant for matrices of linear operators, and the recent version (for the scalar case) in the context of ideal convergence. As byproducts, we obtain characterizations of several matrix classes and a generalization of the classical Hahn--Schur theorem. In the proofs we will use an ideal version of the Banach--Steinhaus theorem which has been recently obtained by De Bondt and Vernaeve in [J.~Math.~Anal.~Appl.~\textbf{495} (2021)].Comment: 25pp, comments are welcom

Diffusions on Wasserstein Spaces

We construct a canonical diffusion process on the space of probability measures over a closed Riemannian manifold, with invariant measure the Dirichlet–Ferguson measure. Together with a brief survey of the relevant literature, we collect several tools from the theory of point processes and of optimal transportation. Firstly, we study the characteristic functional of Dirichlet–Ferguson measures with non-negative finite intensity measure over locally compact Polish spaces. We compute such characteristic functional as a martingale limit of confluent Lauricella hypergeometric functions of type D with diverging arity. Secondly, we study the interplay between the self-conjugate prior property of Dirichlet distributions in Bayesian non-parametrics, the dynamical symmetry algebra of said Lauricella functions and Pólya Enumeration Theory. Further, we provide a new proof of J. Sethuraman’s fixed point characterization of Dirichlet–Ferguson measures, and an understanding of the latter as an integral identity of Mecke- or Georgii–Nguyen–Zessin-type. Thirdly, we prove a Rademacher-type result on the Wasserstein space over a closed Riemannian manifold. Namely, sufficient conditions are given for a probability measure P on the Wasserstein space, so that real-valued Lipschitz functions be P-a.e. differentiable in a suitable sense. Some examples of measures satisfying such conditions are also provided. Finally, we give two constructions of a Markov diffusion process with values in the said Wasserstein space. The process is associated with the Dirichlet integral induced by the Wasserstein gradient and by the Dirichlet–Ferguson measure with intensity the Riemannian volume measure of the base manifold. We study the properties of the process, including its invariant sets, short-time asymptotics for the heat kernel, and a description by means of a stochastic partial differential equation

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum