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.

Keywords:

• Composition operator
• Dirichlet series
• Toeplitz operator

AMS Subject Classifications:

• Primary: 47B33
• Secondary: 30D55
• 46E15

Acknowledgements

The author would like to thank Prof. Maofa Wang for his advice regarding this paper, and thank the referee for the insightful comments.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This research is partially supported by National Science Foundation of China (NSFC) [grant number 11271293].