Soient ${𝒦}_1$ et ${𝒦}_2$ deux espaces de Krein de fonctions analytiques dans le disque unité invariants pour l’opérateur de déplacement à gauche $R_0(R_{0}f\left(z\right)=\left(f\right(z)-f(0\left)\right)/z)$ et soit $A$ un opérateur linéaire continu de ${𝒦}_1$ dans ${𝒦}_2$ dont l’adjoint commute avec $R_0$. Nous étudions les dilatations $B$ de $A$ qui conservent cette propriété de commutation et pour lesquelles les formes hermitiennes définies par $I-A{A}^{*}$ et $I-B{B}^{*}$ ont le même nombre de carrés négatifs. Nous obtenons ainsi une version du théorème de dilatation des commutants d’opérateurs dans le cadre...

In 1999 Nina Zorboska and in 2003 P. S. Bourdon, D. Levi, S. K. Narayan and J. H. Shapiro investigated the essentially normal composition operator $C_{\varphi }$, when $\varphi$ is a linear-fractional self-map of $𝔻$. In this paper first, we investigate the essential normality problem for the operator $T_w{C}_{\varphi }$ on the Hardy space $H^2$, where $w$ is a bounded measurable function on $\partial 𝔻$ which is continuous at each point of $F\left(\varphi \right)$, $\varphi \in 𝒮\left(2\right)$, and $T_w$ is the Toeplitz operator with symbol $w$. Then we use these results and characterize the essentially normal...

On étudie dans ce travail les projections de norme 1 du bidual $E^{\text{'}\text{'}}$ d’un espace de Banach $E$ sur l’image canonique $i_E\left(E\right)$ de $E$ dans $E^{\text{'}\text{'}}$. On montre que dans un certain nombre de cas, il y a unicité de la projection de norme 1. On en déduit des théorèmes d’existence et d’unicité du prédual de $E$. On donne ensuite diverses applications, en particulier aux espaces dont la norme est différentiable sur un ensemble dense et aux espaces ne contenant pas ${\ell }^1\left(N\right)$.

The Newman-Shapiro Isometry Theorem is proved in the case of Segal-Bargmann spaces of entire vector-valued functions (i.e. summable with respect to the Gaussian measure on ℂⁿ). The theorem is applied to find the adjoint of an unbounded Toeplitz operator $T_{\phi }$ with φ being an operator-valued exponential polynomial.

By a (generalized) Fock space we understand a Hilbert space of entire analytic functions in the complex plane C which are square integrable with respect to a weight of the type e-Q(z), where Q(z) is a quadratic form such that tr Q > 0. Each such space is in a natural way associated with an (oriented) circle C in C. We consider the problem of interpolation between two Fock spaces. If C0 and C1 are the corresponding circles, one is led to consider the pencil of circles generated by C0 and C1....

This note explains how the two measures used to define the μ-deformed Segal-Bargmann space are natural and essentially unique structures. As is well known, the density with respect to Lebesgue measure of each of these measures involves a Macdonald function. Our primary result is that these densities are the solution of a system of ordinary differential equations which is naturally associated with this theory. We then solve this system and find the known densities as well as a "spurious" solution...

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...

We develop several methods of realization of scalar product and generalized moment problems. Constructions are made by use of a Hilbertian method or a fixed point method. The constructed solutions are rational fractions and exponentials of polynomials. They are connected to entropy maximization. We give the general form of the maximizing solution. We show how it is deduced from the maximizing solution of the algebraic moment problem.

L. de Branges has originated a viewpoint one of whose repercussions has been the detailed analysis of certain Hilbert spaces of holomorphic functions contained within the Hardy space H2 of the unit disk (...).