Essentially normal multiplication operators on the Dirichlet space (Q1895956)

Let \(D\) be the Dirichlet space of analytic functions \(f(z)= \sum^\infty_{n= 1} a_n z^n\) in the unit disc \(U\) such that \(f(0)= 0\) and \(\sum^\infty_{n= 1} n|a_n|^2< \infty\). An analytic function \(\varphi\) on \(U\) is called a multiplier of \(D\) if \(\varphi D\subset D\). Each multiplier generates a bounded multiplication operator \(M_\varphi\) on \(D\), \(M_\varphi f= \varphi f\) for \(f\in D\). About ten years ago Axler and Shields asked whether \(M^*_\varphi M_\varphi- M_\varphi M^*_\varphi\) is compact for every multiplier \(\varphi\). The paper gives a negative answer to this question. The author shows that the answer is equivalent to the problem of compactness of \(M_{\varphi'}\).