Toeplitzness of composition operators on a Hilbert space of Dirichlet series

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