We derive an integral representation for Herglotz-Nevanlinna functions in two variables, which provides a complete characterization of this class in terms of a real number, two non-negative numbers and a positive measure satisfying certain conditions. Further properties of the class of representing measures are also discussed.

Motivated by reformulating Furstenberg's $\times p,\times q$ conjecture via representations of a crossed product $C^*$-algebra, we show that in a discrete $C^*$-dynamical system $(A,\Gamma)$, the space of (ergodic) $\Gamma$-invariant states on $A$ is homeomorphic to a subspace of (pure) state space of $A\rtimes\Gamma$. Various applications of this in topological dynamical systems and representation theory are obtained. In particular, we prove that the classification of ergodic $\Gamma$-invariant regular Borel probability measures on a compact Hausdorff space $X$ is equivalent to the classification a special type of irreducible representations of $C(X)\rtimes \Gamma$.

We consider the convolution operators in spaces of functions which are holomorphic in a bounded convex domain in ℂ^{$n$}and have a polynomial growth near its boundary. A characterization of the surjectivity of such operators on the class of all domains is given in terms of low bounds of the Laplace transformation of analytic functionals defining the operators.

We show that there are 2^{2^{ℵ_0}} different closed ideals in the Banach algebra L(L_p(0,1)),1<p≠2<∞. This solves a problem in A. Pietsch’s 1978 book “Operator Ideals”. The proof is quite different from other methods of producing closed ideals in the space of bounded operators on a Banach space; in particular, the ideals are not contained in the strictly singular operators and yet do not contain projections onto subspaces that are non-Hilbertian. We give a criterion for a space with an unconditional basis to have 2^{2^{ℵ_0}} closed ideals in terms of the existence of a single operator on the space with some special asymptotic properties. We then show that for 1<q<2 the space Xq of Rosenthal, which is isomorphic to a complemented subspace of L_q(0,1), admits such an operator.

On montre que la fonction maximale centrée de Hardy–Littlewood,$M$, sur les espaces hyperboliques réels $\mathbb{H}^{n} = \mathbb{R}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}}{\raise .17ex\hbox {$\scriptstyle +$}}{\raise .1ex\hbox {$\scriptscriptstyle +$}}{\scriptscriptstyle +}}} \times \mathbb{R}^{n - 1}$ , satisfait l’inégalité de type faible $\| M f \|_{L^{1, \infty}} \leq A (n \log {n}) \| f\|_{1}$ pour toute$f$∈$L$^{1}(ℍ^{$n$}), où$A$>0 est une constante indépendante de la dimension$n$.