On extended eigenvalues and extended eigenvectors of truncated shift (Q2851041)

Let \(H^2\) denote the Hardy space of analytic functions in the open unit disc of the complex plane having square-summable Taylor coefficients at the origin. Let \(S:H^2\longrightarrow H^2\) denote the unilateral shift operator defined by \(Sf(z):=zf(z),~f \in H^2\). For a non-constant inner function \(u\), let \(K_u^2:=H^2\ominus uH^2\). Then it is known that \(K_u^2\) is a proper nontrivial invariant subspace of the adjoint of \(S\). Denote by \(S_u\), the compression of \(S\) to \(K_u^2\). Recall that for a bounded linear operator \(T\) on a Hilbert space \(K\), a complex number \(\lambda\) is called an extended eigenvalue of \(T\) if there is a nonzero bounded linear operator \(X\) on \(K\) such that \(TX=\lambda XT\). Such an \(X\) is referred to as an extended eigenvector for \(\lambda\). The author gives a complete description of the set of extended eigenvectors of \(S_u\) when \(u\) is a Blaschke product.