Spectral properties of generalized Cesàro operators
Let \(g\) be an analytic function on the (open) unit disk \(D\). The generalized Cesàro operator on \({\mathcal H}^2(D)\) with symbol \(g\) is defined by \[ {\mathcal C}_g f (z)=\frac{1}{z}\int_0^z f(t) g(t) \,dt,\quad z\in D. \] For \(g(t)=\frac{1}{1-t}\), \(t\in D\), we obtain the so-called Cesàro operator on \({\mathcal H}^2(D)\). The paper is devoted to the spectral properties of a bounded generalized Cesàro operator \({\mathcal C}_g\) with a rational symbol \(g\) (the spectrum of the Cesàro operator on \({\mathcal H}^2(D)\) has been known for a long time). This means that \(g\) can have only simple poles on the unit circle \(\mathbb T\). The main result of the present paper (Theorem 3.3) characterizes the spectrum of \({\mathcal C}_g\) in terms of the partial fraction decomposition of \(g\) relative to \(\mathbb T\), \[ g(z)=\sum_{i=1}^n \frac{a_i}{1-\beta_i z} + S(z), \] where \(S\) is a rational function analytic in the closed unit disk: namely, the spectrum of \({\mathcal C}_g\) consists of the union of the \(n\) closed disks \(\overline{D (a_i)}\), \(D (a_i)=\{ z: | z-a_i| <| a_i| \), with some possibly isolated points, the set of which is given by \({\mathcal E}=\{ \frac{g(0)}{k} \}_{k=1}^\infty \setminus \cup_{i=1}^n \overline{D (a_i)}\). The essential spectum, \(\sigma_e ({\mathcal C}_g)\), is the union of the \(n\) circles \(\partial D (a_i)\), \(i=1,\ldots, n\), and, moreover, for \(\lambda \notin \sigma_e ({\mathcal C}_g)\), \(\text{ind}\, ({\mathcal C}_g-\lambda I)=-G(\lambda)\), where \(G=\sum_{i=1}^n \chi_{D (a_i)}\). The author uses this result in connection with the BDF theory to show that a generalized Cesàro operator with a rational symbol is unitarily equivalent to a direct sum of multiples of the Cesàro operator, up to a compact perturbation. The case where the poles of the rational function are rationally related is considered in detail. At the end of the paper, some examples are provided, ilustrating cases where the set \({\mathcal E}\) has a finite or a countable number of distinct eigenvalues.
scientific article; zbMATH DE number 2113889