For a pointwise multiplier ϕ of the Hardy-Sobolev space H 2β on the open unit ball B n in C n , we study spectral properties of the multiplication operator M ϕ : H 2 β → H 2 β . In particular, we compute the spectrum and essential spectrum of M ϕ and develop the Fredholm theory for these operators.CAO:

We study the relationship between polynomial approximations in the Bergman space of certain simply connected domains in the complex plane and composition operators on the Dirichlet space of the unit disk. In particular, we characterize when a composition operator on the Dirichlet space has dense range, which settles a problem posed by Joseph Cima in 1976.

Let H be a Hilbert space of holomorphic functions on a bounded domain normalΩ in Cn. Under very mild conditions on normalΩ and H and using the theory of proper holomorphic maps, we present a characterization of holomorphic self‐maps φ:Ω→Ω such that the composition operator Cφ is Fredholm on H.

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