Let $H\left(G_1\right)$ be the set of all holomorphic functions on the domain $G_1.$ Two domains $G_1$ and $G_2$ are called Hadamard-isomorphic if $H\left(G_1\right)$ and $H\left(G_2\right)$ are isomorphic algebras with respect to the Hadamard product. Our main result states that two admissible domains are Hadamard-isomorphic if and only if they are equal.

Let K be an ultraspherical hypergroup associated to a locally compact group G and a spherical projector π and let VN(K) denote the dual of the Fourier algebra A(K) corresponding to K. In this note, invariant means on VN(K) are defined and studied. We show that the set of invariant means on VN(K) is nonempty. Also, we prove that, if H is an open subhypergroup of K, then the number of invariant means on VN(H) is equal to the number of invariant means on VN(K). We also show that a unique topological...

Let $R$ be a hyperbolic Riemann surface, $d_{\chi }$ a harmonic measure supported on the Martin boundary of $R$, and $H^{\infty }\left(d\chi \right)$ the subalgebra of $L^{\infty }\left(d\chi \right)$ consisting of the boundary values of bounded analytic functions on $R$. This paper gives a complete classification of the closed $H^{\infty }\left(d\chi \right)$-submodules of $L^p\left(d\chi \right)$, $1\le p\le \infty$ (weakly${}^{*}$ closed, if $p=\infty$, when $R$ is regular and admits a sufficiently large family of bounded multiplicative analytic functions satisfying an approximation condition. It also gives, as a corollary, a corresponding result for the Hardy...

We considerably improve our earlier results [Ann. Inst. Fourier, 24-4 (1974] concerning Cauchy-Read’s theorems, convergence of Green lines, and the structure of invariant subspaces for a class of hyperbolic Riemann surfaces.

Analogues of the classical Banach-Stone theorem for spaces of continuous functions are studied in the context of the spaces of absolutely continuous functions introduced by Ashton and Doust. We show that if AC(σ₁) is algebra isomorphic to AC(σ₂) then σ₁ is homeomorphic to σ₂. The converse however is false. In a positive direction we show that the converse implication does hold if the sets σ₁ and σ₂ are confined to a restricted collection of compact sets, such as the set of all simple polygons.

Let $\mu$ be a measure on a domain $\Omega$ in ${ℂ}^n$ such that the Bergman space of holomorphic functions in $L^2(\Omega ,\mu )$ possesses a reproducing kernel $K(x,y)$ and $K(x,x)\>0$$\forall x\in \Omega$. The Berezin transform associated to $\mu$ is the integral...