Isomorphic operator algebras and conjugate inner functions
Let \(D\) be the open unit disk of the complex plane and \(m\) the normalized arclength measure on \(\partial U\). If \(\varphi\) is a non constant inner function on \(D\), let \(C_ \varphi\in L(H^ 2)\) be the composition operator \(C_ \varphi(f)=f\circ\varphi\). If \(f\in H^ \infty\), let \(T_ f\) be the analytic Toeplitz operator \(T_ f\in L(H^ 2)\), \(T_ f(h)=fh\). Let \(A_ \varphi\) denote the norm closed algebra generated by \(C_ \varphi\) and \(\{T_ f\): \(f\in H^ \infty\}\). The main result is that, given \(\varphi\) and \(\psi\) nonconstant nonperiodic inner functions, then there exists an analytic homeomorphism \(\tau\) of \(D\) such that \(\tau\circ\psi=\varphi\circ\tau\) if and only if \(A_ \varphi\) and \(A_ \psi\) are isomorphic.
scientific article; zbMATH DE number 93891