The commutant of a certain compression

Abstract:

Let $G$ be any bounded region in the complex plane and $K \subset G$ be a simple compact arc of class ${C^1}$. Let ${A^2}(G\backslash K)$ (resp. ${A^2}(G)$) be the Bergman space on $G\backslash K$ (resp. $G$). Let $S$ be the operator multiplication by $z$ on ${A^2}(G\backslash K)$ and $C = {P_\mathcal {N}}S{|_\mathcal {N}}$ be the compression of $S$ to the semi-invariant subspace $\mathcal {N} = {A^2}(G\backslash K) \ominus {A^2}(G)$. We show that the commutant of ${C^{\ast }}$ is the set of all operators of the form ${A^{ - 1}}{M_h}A$, where $h$ is a multiplier on a certain Sobolev space of functions on $K$ and $(Af)(w) = \int _G {f(z){{(\overline z - \overline w )}^{ - 1}}dA(z)(w \in K)}$. We also use multiplier theory in fractional order Sobolev spaces to obtain further information about $C$.