On the commutant of certain multiplication operators on spaces of analytic functions (Q1864677)

Let \(\mathcal{M}(B)\) be the space of multipliers in the Banach space \(B\) of analytic functions defined on the open unit disk \(D\) and let \(M_\varphi (f)=\varphi f\) denote the multiplication operator on \(B\), where \(\varphi\) is a complex valued function on \(D\). The author characterizes the commutant of \(M_{z^n}, n>1\). He shows that if the operator \(S\) commutes with the operator \(M_{z^2}\) and \(SM_{z^{2k+1}}-M_{z^{2k+1}}S\) is compact for some \(k\in N\) then there exists some \(\varphi \in \mathcal{M}(B)\) such that \(S=M_\varphi\). The analogous assertion \(S=M_\varphi\) under some additional assumptions also holds if \(S\) commutes with the operator \(M_{z^n}, n>1\) on a functional Hilbert spaces of analytic functions and \(SM_{z}-M_{z}S\) is a compact operator.