arXiv subjects

Benedict Bauer

Publications and source records attributed to Benedict Bauer.

3 recordsLinked to original sources

arXiv2023

A characterization of real matrix semigroups

We characterize all real matrix semigroups satisfying a mild boundedness assumption, without assuming continuity. Besides the continuous solutions of the semigroup functional equation, we give a description of solutions arising from non-measurable solutions of Cauchy's functional equation. To do so, we discuss the primary decomposition and the Jordan-Chevalley decomposition of a matrix semigroup. Our motivation stems from a characterization of all multi-dimensional self-similar Gaussian Markov processes, which is given in a companion paper.

math.RA↗

arXiv2023

The Fan-Taussky-Todd inequalities and the Lumer-Phillips theorem

We argue that a classical inequality due to Fan, Taussky and Todd (1955) is equivalent to the dissipativity of a Jordan block. As the latter can be characterised via the zeros of Chebyshev polynomials, we obtain a short new proof of the inequality. Three other inequalities of Fan-Taussky-Todd are reproven similarly. By the Lumer-Phillips theorem, the semigroup defined by the Jordan block is contractive. This yields new extensions of the classical Fan-Taussky-Todd inequalities. As an application, we give an estimate for the partial sums of a Bessel function.

math.CA↗

arXiv2020

Self-similar Gaussian Markov processes

We characterize all multi-dimensional real self-similar Gaussian Markov processes. Three types of covariance matrix functions occur: white-noise type functions, covariances that can be expressed by continuous matrix semigroups, and covariances based on non-continuous solutions of Cauchy's functional equation. Characterizing the latter requires us to develop some results on the representation theory of non-continuous matrix semigroups, which are presented in a companion paper. In dimension one, besides white noise, the self-similar Gaussian Markov processes reduce to a two-parameter family of time-changed Brownian motions. This observation simplifies several proofs of non-Markovianity of concrete processes found in the literature.

math.PR↗