In this paper, we give an integral representation for the boundary values of derivatives of functions of the de Branges–Rovnyak spaces $\mathscr{H}\left(b\right)$, where $b$ is in the unit ball of $H^{\infty }\left({ℂ}_{+}\right)$. In particular, we generalize a result of Ahern–Clark obtained for functions of the model spaces $K_b$, where $b$ is an inner function. Using hypergeometric series, we obtain a nontrivial formula of combinatorics for sums of binomial coefficients. Then we apply this formula to show the norm convergence of reproducing kernel $k_{\omega ,n}^b$ of evaluation...

Soit $f$ une transformée de Stieltjes. Notant $H_f$ un prolongement de la fonction $f\left(z^{-1}\right)$ à $(\mathbf{C}\setminus {\mathbf{R}}^{*}\cup \{\infty \left\}\right)$, on définit, pour tout espace de Banach $X$ et pour tout opérateur $V$ sur $X$ qui soit de domaine dense, fermé, d’ensemble résolvant contenant ${\mathbf{R}}^{*}$ et qui vérifie ${sup}_{\lambda \>0}\parallel (I+\lambda V{)}^{-1}\parallel \<\infty$, un opérateur $H_f\left(V\right)$ qui est un opérateur sur $X$ de même nature que $V$. On montre que l’on a ${\sigma }^e[H_f\left(V\right)]=H_f[{\sigma }^e\left(V\right)]$ (où ${\sigma }^e$ désigne le spectre étendu). En outre, l’opération $H_f$ a d’excellentes propriétés de stabilité. En particulier, si $f\ne 0$ et si $V$ est un potentiel abstrait, $H_f\left(V\right)$ est un potentiel...

Given two n-tuples $a=(a_1,...,a_n)$ and $b=(b_1,...,b_n)$ of bounded linear operators on a Hilbert space the question of when there exists an elementary operator E such that $E{a}_j=b_j$ for all j =1,...,n, is studied. The analogous question for left multiplications (instead of elementary operators) is answered in any C*-algebra A, as a consequence of the characterization of closed left A-submodules in $A^n$.

The behavior of the essential spectrum and the essential norm under (complex/real) interpolation is investigated. We extend an example of Albrecht and Müller for the spectrum by showing that in complex interpolation the essential spectrum ${\sigma }_e\left(S_{\left[\theta \right]}\right)$ of an interpolated operator is also in general a discontinuous map of the parameter θ. We discuss the logarithmic convexity (up to a multiplicative constant) of the essential norm under real interpolation, and show that this holds provided certain compact approximation...

A new criterion for the existence of an invariant distribution for Markov operators is presented. Moreover, it is also shown that the unique invariant distribution of an iterated function system is singular with respect to the Hausdorff measure.

A new sufficient condition for the existence of an invariant measure for Markov operators defined on Polish spaces is presented. This criterion is applied to iterated function systems.