A relative of the shift operator (Q1187375)

For a positive number \(\alpha\), define \(E_ \alpha\) by \[ E_ \alpha=\{\lambda\in \mathbb{C}:\;| \lambda-1|+|\lambda +1|\leq \alpha+\alpha^{-1}\}. \] For \(\alpha\neq 1\), \(E_ \alpha\) is a closed elliptic disc. If \(U\) is the unilateral shift on a Hilbert space \(H\), then for any complex number \(\alpha\) define \(U_ \alpha\) by \(U_ \alpha={1\over 2}(\alpha U+\alpha^{-1} U^*)\), where \(U^*\) is the adjoint of \(U\). In this article the author proves the following Theorem: Let \(\alpha\) be a positive number then \(\sigma(U_ \alpha)=E_ \alpha\), and for \(\lambda\in E_ \alpha^ e\), the resolvent of \(U_ \alpha\) can be written as \[ R(\lambda,U_ \alpha)=2\xi_ 2 R\left({{\xi_ 2} \over \alpha},U\right) R(\alpha\xi_ 2,U^*), \] where \(|\xi_ 2| >1\), denotes the root of the equation \(\xi^ 2-2\lambda\xi+1=0\). Moreover, \(\sigma_ e(U_ \alpha)=\partial E_ \alpha\), \(\sigma_ p(U_ \alpha)=\text{int }E_ \alpha\), \(\sigma_ r(U_ \alpha)=\emptyset\), for \(\alpha<1\), \(\sigma_ r(U_ \alpha)=\text{int }E_ \alpha\), \(\sigma_ p(U_ \alpha)=\emptyset\), for \(\alpha>1\), where \(\sigma(U_ \alpha)\), \(\sigma_ p(U_ \alpha)\), \(\sigma_ e(U_ \alpha)\), and \(\sigma_ r(U_ \alpha)\) are respectively the spectrum, point spectrum, continuous spectrum and residual spectrum of \(U_ \alpha\).