Sampling sets and closed range composition operators on the Bloch space

Abstract

We give a necessary and sufficient condition for a composition operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript phi"> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>ϕ</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">C_{\phi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the Bloch space to have closed range. We show that when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi>ϕ</mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is univalent, it is sufficient to consider the action of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript phi"> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>ϕ</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">C_{\phi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the set of Möbius transforms. In this case the closed range property is equivalent to a specific sampling set satisfying the reverse Carleson condition.