Toeplitzness of composition operators on a Hilbert space of Dirichlet series

Abstract

In this paper, we investigate the -Toeplitzness and asymptotic -Toeplitzness of composition operators and their adjoints on the Hilbert space of Dirichlet series with square summable coefficients. The operator is a shift specific to that space. We obtain characterizations of -Toeplitz and -asymptotically Toeplitz composition operators, where the asymptotic concepts are relative to the uniform, strong, respectively, and weak operator topologies. A characterization of strong asymptotic -Toeplitzness of adjoints of composition operators is also obtained.