Complex symmetric composition operators on weighted Hardy spaces

Abstract

Let <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">\varphi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an analytic self-map of the open unit disk <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We study the complex symmetry of composition operators <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:mi>φ</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">C_\varphi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on weighted Hardy spaces induced by a bounded sequence. For any analytic self-map of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that is not an elliptic automorphism, we establish that if <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_{\varphi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is complex symmetric, then either <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi left-parenthesis 0 right-parenthesis equals 0"> <mml:semantics> <mml:mrow> <mml:mi>φ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\varphi (0)=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <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">\varphi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is linear. In the case of weighted Bergman spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript alpha Superscript 2"> <mml:semantics> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>α</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">A^{2}_{\alpha }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we find the non-automorphic linear fractional symbols <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">\varphi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <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_{\varphi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is complex symmetric.