For \(\phi \) and \(\psi \) inner functions that fix the origin on the open unit disk \(\mathbb {D}\) in the complex plane, we consider the question of whether the associated linear operator \(C_{\phi }^*C_{\psi }\) can be compact or finite-rank on \(H^2(\mathbb {D})\) . We show that \(C_{\phi }^*C_{\psi }\) cannot be rank-one when \(\phi \) has purely atomic Aleksandrov–Clark measure and \(\psi \) extends continuously to the boundary of \(\mathbb {D}\) . When \(\phi \) and \(\psi \) are finite Blaschke products each with two distinct factors, we show \(C_{\phi }^*C_{\psi }\) cannot be compact. Finally, following work of Cowen and MacCluer, we characterize the range of \(C_{\phi }^*\) when \(\phi \) is a Blaschke product.
