Compact differences of weighted composition operators
Bin Liu, Jouni Rättyä
Abstract
Abstract Compact differences of two weighted composition operators acting from the weighted Bergman space $$A^p_{\omega }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mi>ω</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> </mml:math> to another weighted Bergman space $$A^q_{\nu }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mi>ν</mml:mi> <mml:mi>q</mml:mi> </mml:msubsup> </mml:math> , where $$0<p\le q<\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo><</mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>q</mml:mi> <mml:mo><</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> and $$\omega ,\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ν</mml:mi> </mml:mrow> </mml:math> belong to the class $${\mathcal {D}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proof a new description of q -Carleson measures for $$A^p_{\omega }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mi>ω</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> </mml:math> , with $$\omega \in {\mathcal {D}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>D</mml:mi> </mml:mrow> </mml:math> , in terms of pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q -Carleson measures for the classical weighted Bergman space $$A^p_{\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mi>α</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> </mml:math> with $$-1<\alpha <\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo><</mml:mo> <mml:mi>α</mml:mi> <mml:mo><</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> to the setting of doubling weights.