Adjoints of linear fractional composition operators on the Dirichlet space
Let \(T\) be an open unit disc on the complex plane and \(\mathcal{D}\) the Dirichlet space of functions analytic on \(T\) equipped with the norm defined by \[ \| f\| ^2=| f(0)| ^2+\int\limits_T| f^\prime(z)| ^2\,dA(z). \] The authors study the adjoints of the composition operators \(C_\varphi f= f\circ \varphi\) with a fractional linear function \(\varphi(z)=\frac{az+b}{cz+d}\), \(ad-bc \neq 0\), \(\varphi(T)\subset T\). It is shown that these adjoints may be expressed as a fractional linear operator plus a rank two operator acting on \(\mathcal{D}\) itself. The authors observe that the proofs are different from those known for the Hardy spaces. The key point in that difference is due to the fact that many fractional linear composition operators are normal on Dirichlet spaces while \(C_\varphi\) is normal on Hardy spaces if and only if \(\varphi(z)=\lambda z\), \(| \lambda| \leq 1\). As an application, there is given an explicit formula for the norm of a fractional linear composition operator on the Dirichlet spaces modulo constant functions. As another application, the computation of the spectrum and essential spectrum is considered.
scientific article; zbMATH DE number 2001051