On Hankel Operators Associated with Markov Functions

Abstract

In this paper some questions related to Hankel operators associated with Markov functions are considered. Let G be a bounded multiply connected domain with a boundary Γ consisting of closed analytic Jordan curves. We assume that G is symmetric with respect to the real axis. Let µ be a positive Borel measure with the support supp µ = E ⊂ R, E ⊂ G. We investigate a connection between the Hankel operator A _f constructed from the Markov function

$$ f\left( z \right) = \frac{1}{{2\pi i}}\int_{E} {\frac{{d\mu \left( x \right)}}{{z - x}}} $$

and the embedding operator J: E ₂ (G) → L ₂ (μ,E), where E ₂ (G) is the Smirnov class of functions analytic on G. Moreover, in the case when G is the open unit disk we state results characterizing the rate of decrease of the sequence of singular numbers of the Hankel operator A _f constructed from the Markov function with the measure μ satisfying the Szegö condition: supp μ = [a, b] ⊂ (-1,1) and

$$ \int_a^b {\frac{{\log \left( {d\mu /dx} \right)}}{{\sqrt {\left( {x - a} \right)\left( {b - x} \right)} }}dx > - \infty .} $$