Weighted holomorphic Dirichlet series and composition operators with polynomial symbols

Authors

  • Emmanuel Fricain
  • Camille Mau

DOI:

https://doi.org/10.7146/math.scand.a-129686

Abstract

In this paper, we introduce a general class of weighted spaces of holomorphic Dirichlet series (with real frequencies) analytic in some half-plane and study composition operators on these spaces. In the particular case when the symbol inducing the composition operator is an affine function, we give criteria for boundedness and compactness. We also study the cyclicity property and as a byproduct give a characterization so that the direct sum of the identity plus a weighted forward shift operator on $\ell^2$ is cyclic.

References

Apostol, T. M., Modular functions and Dirichlet Series in Number Theory, Springer-Verlag, New York, 1990. https://doi.org/10.1007/978-1-4612-0999-7

Bayart, F., Queffélec, H., Seip, K., Approximation numbers of composition operators on $H^p$ spaces of Dirichlet series, Ann. Inst. Fourier (Grenoble) 66 (2016), no. 2, 551–588. http://aif.cedram.org/item?id=AIF_2016__66_2_551_0

Bayart, F., Hardy spaces of Dirichlet series and their composition operators, Monatsh. Math. 136 (2002), no. 3, 203–236. https://doi.org/10.1007/s00605-002-0470-7

Cowen, C., MacCluer, B., Composition operators on spaces of analytic functions, Studies in Advances in Mathematics, CRC Press, Boca Raton, FL, 1995.

Doan, M. L., Khoi, L. H., Hilbert spaces of entire functions and composition operators, Complex Anal. Oper. Theory 10 (2016), no. 1, 213–230. https://doi.org/10.1007/s11785-015-0497-0

Doan, M. L., Khoi, L. H., Closed range and cyclicity of composition operators on Hilbert space of entire functions, Complex Var. Elliptic Equ. 63 (2018), no. 11, 1558–1569. https://doi.org/10.1080/17476933.2017.1391799

Doan, M. L., Khoi, L. H., Complete characterization of bounded composition operators on the general weighted Hilbert spaces of entire Dirichlet series, North-West. Eur. J. Math. 6 (2020), 91–106.

Doan, M. L., Mau, C., Khoi, L. H., Complex symmetry of composition operators on Hilbert spaces of entire Dirichlet series, Vietnam J. Math. 47 (2019), no. 2, 443–460. https://doi.org/10.1007/s10013-018-00330-6

Garcia, S. R., Putinar, M., Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 (2006), no. 3, 1285–1315. https://doi.org/10.1090/S0002-9947-05-03742-6

Garcia, S. R., Wogen, W., Some new classes of complex symmetric operators, Trans. Amer. Math. Soc. 362 (2010), no. 11, 6065–6077. https://doi.org/10.1090/S0002-9947-2010-05068-8

Gordon, J., Hedenmalm, H., The composition operators on the space of Dirichlet series with square summable coefficients, Michigan Math. J. 46 (1999), no. 2, 313–329. https://doi.org/10.1307/mmj/1030132413

Halmos, P., A Hilbert space problem book, Second edition. Springer-Verlag, New York-Berlin, 1982.

Hardy, G. H., Riesz, M., The general theory of Dirichlet series, Stechert-Hafner, Inc., New York, 1964.

Hedenmalm, H., Dirichlet series and functional analysis. The Legacy of Niels Henrik Abel, 673–684, Springer-Verlag, 2004.

Hilden, H., Wallen, L., Some cyclic and non-cyclic vectors of certain operators, Indiana Univ. Math. J. 23 (1973/74), 557–565. https://doi.org/10.1512/iumj.1974.23.23046

Hou, X., Khoi, L. H., Some properties of composition operators on entire Dirichlet series with real frequencies, Compt. Rend. Math. Acad. Sci. Paris 350 (2012), no. 3–4, 149–152. https://doi.org/10.1016/j.crma.2012.01.023

Hou, X., Hu, B., Khoi, L. H., Hilbert spaces of entire Dirichlet series and composition operators, J. Math. Anal. Appl. 401 (2013), no. 1, 416–429. https://doi.org/10.1016/j.jmaa.2012.12.036

Hu, B., Khoi, L. H., Zhao, R., Topological structure of the spaces of composition operators on Hilbert spaces of Dirichlet series, Z. Anal. Anwend. 35 (2016), no. 3, 267–284. https://doi.org/10.4171/ZAA/1565

Hu, B., Khoi, L. H., Zhu, K. Frames and operators in Schatten classes, Houston J. Math. 41 (2015), no. 4, 1191–1219.

Queffélec H., Seip, K., Approximation numbers of composition operators on the $H^2$ space of Dirichlet series, J. Funct. Anal. 268 (2015), no.6, 1612–1648. https://doi.org/10.1016/j.jfa.2014.11.022

Queffélec H., Espaces de séries de Dirichlet et leurs opérateurs de composition, Ann. Math. Blaise Pascal 22 (2015), no. S2, 267–344. http://ambp.cedram.org/item?id=AMBP_2015__22_2_267_0

Seubert, S. M., Cyclic vectors on shift coinvariant subspaces, Rocky Mountain J. Math. 24 (1994), no. 2, 719–727. https://doi.org/10.1216/rmjm/1181072429

Shapiro, J., Composition operators and classical function theory, Springer-Verlag, New York, 1993. https://doi.org/10.1007/978-1-4612-0887-7

Valiron, G., Théorie générale des séries de Dirichlet, Mémorial des Sciences Mathématiques, Fascicule 17 (1926).

Wang, M., Yao, X., Some properties of composition operators on Hilbert spaces of Dirichlet series, Complex Var. Elliptic Equ. 60 (2015), no. 7, 992–1004. https://doi.org/10.1080/17476933.2014.988617

Yu, J. R., Ding, X. Q., Tian, F. J., On the distribution of values of Dirichlet series and random Dirichlet series, Press in Wuhan Univ., Wuhan, China, 2004.

Published

2022-02-24

How to Cite

Fricain, E., & Mau, C. (2022). Weighted holomorphic Dirichlet series and composition operators with polynomial symbols. MATHEMATICA SCANDINAVICA, 128(1). https://doi.org/10.7146/math.scand.a-129686