On the rotational invariance for the essential spectrum of \(\lambda\)-Toeplitz operators
Let \({\mathcal H}\) be a separable Hilbert space with an orthonormal basis \(\{e_n\}_{n=0}^\infty\) and \(\lambda\in\overline{\mathbb D}=\{z\in{\mathbb C}:|z|\leq 1\}\). A bounded operator \(T\) is called a \(\lambda\)-Toeplitz if \(\langle Te_{m+1},e_{n+1}\rangle=\lambda\langle Te_m,e_n\rangle\) for all \(m,n\geq 0\). Let \(\sigma(T)\) and \(\sigma_e(T)\) denote the spectrum and the essential spectrum of a bounded linear operator \(T\) on \({\mathcal H}\). For \(\lambda\in{\mathbb T}\) and \(\varphi\in L^2({\mathbb T})\) given by its Fourier series \(\sum_{n=-\infty}^\infty a_ne^{in\theta}\), let \(T_{\lambda,\varphi}\) be the \(\lambda\)-Toeplitz operator on the Hardy space \(H^2({\mathbb T})\). Consider the function \(\varphi_{\overline{\lambda},+}\) given by its Fourier series \(\sum_{n=-\infty}^\infty b_ne^{in\theta}\) with \(b_n=\overline{\lambda}^na_n\) if \(n\geq 0\) and \(b_n=a_n\) if \(n<0\). It is shown that if \(\varphi_{\overline{\lambda},+}\in L^\infty({\mathbb T})\), then \(\sigma_e(T_{\lambda,\varphi})=\lambda\sigma_e(T_{\lambda,\varphi})\) (``rotational invariance''). If \(\lambda\) is not of the form \(e^{2i\pi(p/q)}\), then \(\sigma_e(T_{\lambda,\varphi})\) is circular. If \(\lambda=e^{2i\pi(p/q)}\) with the rational number \(p/q\) in the lowest terms, then \(\lambda\) is said to be of order \(q\). If \(\lambda\) has order \(q\) and \(\varphi_{\overline{\lambda},+}\in C({\mathbb T})\), then \[ \sigma_e(T_{\lambda,\varphi})=\{\mu\in{\mathbb C}:\mu^q\in\Phi_{\overline{\lambda}+}({\mathbb T})\}, \] where \[ \Phi_{\overline{\lambda}+}=\prod_{j=0}^{q-1}\varphi_{\overline{\lambda},+}\circ\tau^j \] and \(\tau(e^{i\theta})=\overline{\lambda}e^{i\theta}\). Moreover, the Fredholm index of \(T_{\lambda,\varphi}\) is equal to \(-q^{-1}\operatorname{wn}(\Phi_{\overline{\lambda}+}-\mu^q)\), where \(\text{wn}(\cdot)\) denotes the winding number. Finally, if \(\lambda\) has order \(q\) and \(\varphi_{\overline{\lambda},+}\in H^\infty({\mathbb T})\), then \[ \sigma(T_{\lambda,\varphi})=\{\mu\in{\mathbb C}:\mu^q\in\operatorname{cl}(\Phi_{\overline{\lambda}+}({\mathbb D}))\}. \]
scientific article; zbMATH DE number 6420279
scientific article