On a proposed characterization of Schatten-class composition operators

Source abstract

For an analytic function φ \varphi which maps the open unit disc D D to itself, let C φ C_{\varphi } be the operator of composition with φ \varphi on the Bergman space L a 2 ( D , d A ) L^{2}_{a}(D,dA) . It has been a longstanding problem to determine whether or not the membership of C φ C_{\varphi } in the Schatten class C p {\mathcal {C}}_{p} , 1 > p > ∞ 1 > p > \infty , is equivalent to the condition that the function z ↦ { ( 1 − | z | 2 ) / ( 1 − | φ ( z ) | 2 ) } p z \mapsto \{(1-|z|^{2})/(1-|\varphi (z)|^{2})\}^{p} has a finite integral with respect to the Möbius-invariant measure d λ ( z ) = ( 1 − | z | 2 ) − 2 d A ( z ) d\lambda (z) = (1-|z|^{2})^{-2}dA(z) on D D . We show that the answer is negative when 2 > p > ∞ 2 > p > \infty .