Pommiez operator in spaces of analytic functions in the circle (Q2761518)

Let \(A_{R}\) \((0<R\leq\infty)\) be a space of all single-valued and analytic functions in \(\{z\in{\mathbb C}:|z|<R\}\) with topology of compact convergence. The operator \((\Delta f)(z)=(f(z)-f(0))z^{-1}\) is called Pommiez operator. We denote by \(L(A_{R})\) the space of all linear continuous operators in \(A_{R}\). The author proves that the operator \(T\) belongs to the set \(L(A_{R})\) and satisfies the condition NEWLINE\[NEWLINET(fg)(z)=f(z)(Tg)(z)+g(0)(Tf)(z),\;\forall f(z),g(z)\in A_{R}NEWLINE\]NEWLINE if and only if \(T\) has a form \(T=U_{\varphi}\Delta\), where \((U_{\varphi}f)(z)=\varphi(z)f(z)\), \(\varphi(z)\) is fixed function from \(A_{R}\). Operators of infinite order with respect to \(\Delta\), the commutant of the operator \(\Delta^{p},\;p\in N\), the operator \(\Delta\) and rational functions are studied. Conditions of equivalence of the operator NEWLINE\[NEWLINE\Delta^{p}+\varphi_{1}(z)\Delta^{p-1}+\ldots+\varphi_{p-1}(z)\Delta+\varphi_{p}E,\;p\in N,\;\varphi_{k}(z)\in A_{R},\;k=1,2,\ldots,pNEWLINE\]NEWLINEand the operator \(\Delta^{p}\) are proposed.