The article introduces and studies Hausdorff–Berezin operators on the unit ball in a complex space. These operators are a natural generalization of the Berezin transform. In addition, the class of such operators contains, for example, the invariant Green potential, and some other operators of complex analysis. Sufficient and necessary conditions for boundedness in the space of p – integrable functions with Haar measure (invariant with respect to involutive automorphisms of the unit ball) are given. We also provide results on compactness of Hausdorff–Berezin operators in Lebesgue spaces on the unit ball. Such operators have previously been introduced and studied in the context of the unit disc in the complex plane. Present work is a natural continuation of these studies.

We construct a class of non-reversible Metropolis kernels as a multivariate extension of the guided-walk kernel proposed by Gustafson (Statist. Comput. 8, 1998). The main idea of our method is to introduce a projection that maps a state space to a totally ordered group. By using Haar measure, we construct a novel Markov kernel termed the Haar mixture kernel, which is of interest in its own right. This is achieved by inducing a topological structure to the totally ordered group. Our proposed method, the $\Delta$-guided Metropolis–Haar kernel, is constructed by using the Haar mixture kernel as a proposal kernel. The proposed non-reversible kernel is at least 10 times better than the random-walk Metropolis kernel and Hamiltonian Monte Carlo kernel for the logistic regression and a discretely observed stochastic process in terms of effective sample size per second.

For any family of $N\,\times \,N$ random matrices ${{\left( {{\text{A}}_{k}} \right)}_{k\in K}}$ that is invariant, in law, under unitary conjugation, we give general sufficient conditions for central limit theorems for random variables of the type $\text{Tr}\left( {{\text{A}}_{k}}\text{M} \right)$, where the matrix $\text{M}$ is deterministic (such random variables include, for example, the normalized matrix entries of ${{\text{A}}_{k}}$). A consequence is the asymptotic independence of the projection of the matrices ${{\text{A}}_{k}}$ onto the subspace of null trace matrices from their projections onto the orthogonal of this subspace. These results are used to study the asymptotic behavior of the outliers of a spiked elliptic random matrix. More precisely, we show that the fluctuations of these outliers around their limits can have various rates of convergence, depending on the Jordan Canonical Form of the additive perturbation. Also, some correlations can arise between outliers at a macroscopic distance from each other.These phenomena have already been observed with random matrices from the Single Ring Theorem.

The century-old extremal problem, solved by Carathéodory and Fejér, concerns a non-negative trigonometric polynomial $T(t) = a_0 + \sum\nolimits_{k = 1}^n {a_k} \cos (2\pi kt) + b_k\sin (2\pi kt){\ge}0$, normalized by a0=1, where the quantity to be maximized is the coefficient a1 of cos (2π t). Carathéodory and Fejér found that for any given degree n, the maximum is 2 cos(π/n+2). In the complex exponential form, the coefficient sequence (ck) ⊂ ℂ will be supported in [−n, n] and normalized by c0=1. Reformulating, non-negativity of T translates to positive definiteness of the sequence (ck), and the extremal problem becomes a maximization problem for the value at 1 of a normalized positive definite function c: ℤ → ℂ, supported in [−n, n]. Boas and Kac, Arestov, Berdysheva and Berens, Kolountzakis and Révész and, recently, Krenedits and Révész investigated the problem in increasing generality, reaching analogous results for all locally compact abelian groups. We prove an extension to all the known results in not necessarily commutative locally compact groups.

In this paperwe present a fixed point property for amenable hypergroups that is analogous to Rickert’s fixed point theorem for semigroups. It equates the existence of a left invariant mean on the space of weakly right uniformly continuous functions to the existence of a fixed point for any action of the hypergroup. Using this fixed point property, certain hypergroups are shown to have a left Haar measure.

In a recent article [K. H. Hofmann and F. G. Russo, ‘The probability that $x$ and $y$ commute in a compact group’, Math. Proc. Cambridge Phil Soc., to appear] we calculated for a compact group $G$ the probability $d(G)$ that two randomly selected elements $x, y\in G$ satisfy $xy=yx$, and we discussed the remarkable consequences on the structure of $G$ which follow from the assumption that $d(G)$ is positive. In this note we consider two natural numbers $m$ and $n$ and the probability $d_{m,n}(G)$ that for two randomly selected elements $x, y\in G$ the relation $x^my^n=y^nx^m$ holds. The situation is more complicated whenever $n,m\gt 1$. If $G$ is a compact Lie group and if its identity component $G_0$ is abelian, then it follows readily that $d_{m,n}(G)$ is positive. We show here that the following condition suffices for the converse to hold in an arbitrary compact group $G$: for any nonopen closed subgroup $H$ of $G$, the sets $\{g\in G: g^k\in H\}$ for both $k=m$ and $k=n$ have Haar measure $0$. Indeed, we show that if a compact group $G$ satisfies this condition and if $d_{m,n}(G)\gt 0$, then the identity component of $G$is abelian.

A control system is said to be finite if the Lie algebra generated by its vector fieldsis finite dimensional. Sufficient conditions for such a system on a compact manifold to becontrollable are stated in terms of its Lie algebra. The proofs make use of theequivalence theorem of [Ph. Jouan, ESAIM: COCV 16 (2010)956–973]. and of the existence of an invariant measure on certain compact homogeneousspaces.

Let M be a random matrix chosen from Haar measure on the unitary group Un. Let Z = X + iY be a standard complex normal random variable with X and Y independent, mean 0 and variance ½ normal variables. We show that for j = 1, 2, …, Tr(Mj) are independent and distributed as √jZ asymptotically as n →∞. This result is used to study the set of eigenvalues of M. Similar results are given for the orthogonal and symplectic and symmetric groups.

Compact right topological groups arise naturally as the enveloping semigroups of distal flows. Recently, John Pym and the author established the existence of Haar measure μ on such groups, which invites the consideration of the regular representations. We start here by characterizing the continuous representations of a compact right topological group G, and are led to the conclusion that the right regular representation r is not continuous (unless G is topological). The domain of the left regular representation l is generally taken to be the topological centre

or a tractable subgroup of it, furnished with a topology stronger than the relative topology from G (the goals being to have l both defined and continuous). An analysis of l and r on H = L2(G) for some non-topological compact right topological groups G shows, among other things, that:

1. (i) for the simplest (perhaps) G generated by ℤ, (l, H) decomposes into one copy of each irreducible representation of ℤ and c copies of the regular representation.

2. (ii) for the simplest (perhaps) G generated by the euclidean group of the plane , (l, H) decomposes into one copy of each of the continuous one-dimensional representations of and c copies of each continuous irreducible representation Ua,a > 0.

3. (iii) when Λ(G) is not dense in G, it can seem very reasonable to regard r as a continuous representation of a related compact topological group, and also, G can be almost completely "lost" in the measure space (G, μ).

We find a ‘natural' measure on the set of convex infinite cylinders touching a convex body K in En. Using the curvature measures, we can calculate the probability that a convex infinite cylinder and a convex body K are tangent in some previously determined regions. As a particular case we obtain a measure on all q-planes touching K.