Hermitian composition operators on Hardy-Smirnov spaces

Let Ω be an open simply connected proper subset of the complex plane and φ an analytic self map of Ω. If f is in the Hardy-Smirnov space defined on Ω, then the operator that takes f to f º φ is a composition operator. We show that for any Ω, analytic self maps that induce bounded Hermitian composition operators are of the form Φ(w) = aw + b where a is a real number. For ceratin Ω, we completely describe values of a and b that induce bounded Hermitian composition operators.

On estimations of solutions of one convolution type equation

We consider a convolution type equation in a semi-strip for the functions belonging to the Hardy-Smirnov space. Estimations of solutions are obtained in terms of analytic continuation.