In this paper, we study the complex symmetric $C_0$-semigroups of weighted composition operators $W_{\psi, \varphi}$ on the weighted Hardy spaces $H_\gamma$ of the unit disk $\mathbb{D}$. It is well-known that there are only two classes of weighted composition conjugations $\mathcal{A}_{u, v}$ on $H_\gamma(\mathbb{D})$: either $\mathcal{C}_1$ or $\mathcal{C}_2$. We completely characterize $\mathcal{C}_1$-symmetric ($\mathcal{C}_2$-symmetric) $C_0$-semigroups of weighted composition operators $W_{\psi, \varphi}$ on $H_\gamma(\mathbb{D})$.}
Xiaohe Hu
. COMPLEX SYMMETRIC $C_0$-SEMIGROUPS ON THE WEIGHTED HARDY SPACES $H_\gamma(\mathbb{D})$*[J]. Acta mathematica scientia, Series B, 2023
, 43(5)
: 2121
-2132
.
DOI: 10.1007/s10473-023-0512-x
[1] Abate M. The infinitesimal generators of semigroups of holomorphic maps. Ann Mat Pura Appl, 1992, 161: 167-180
[2] Aizenberg L, Shoikhet D. Boundary behavior of semigroups of holomorphic mappings on the unit ball in $\mathbb{C}_n$. Complex Var Theory Appl, 2002, 47: 109-121
[3] Bisi C, Bracci F. Linear fractional maps of the unit ball: A geometric study. Adv Math, 2002, 167: 265-287
[4] Bracci F, Contreras M D, Díaz-Madrigal S. Classification of semigroups of linear fractional maps in the unit ball. Adv Math, 2007, 208: 318-350
[5] Bracci F, Contreras M D, Díaz-Madrigal S. Infinitesimal generators associated with semigroups of linear fractional maps. J Anal Math, 2007, 102: 119-142
[6] Berkson E, Porta H. Semigroups of analytic functions and composition operators. Michigan Math J, 1978, 25: 111-115
[7] Cowen C C, MacCluer B D. Composition Operators on Spaces of Analytic Functions. Boca Raton, FL: CRC Press, 1995
[8] Davies, E B.One-Parameter Semigroups. London, New York: Academic Press, 1980
[9] Engel K J, Nagel R.One-Parameter Semigroups for Linear Evolution Equations. New York: Springer-Verlag, 2000
[10] Goldstein J A.Semigroups of Operators and Applications. New York: Oxford University Press, 1985
[11] Hai P V, Khoi L H. Complex symmetric $C_0$-semigroups on the Fock space. J Math Anal Appl, 2017, 445: 1367-1389
[12] Han K K, Wang M F. Complex symmetric $C_0$-semigroups on $A^2(\mathbb{C}_+)$. Acta Math Sin, Engl Ser, 2020, 36v: 1171-1182
[13] Jury M. Norms and spectral radii of linear fractional composition operators on the ball. J Funct Anal, 2008, 254: 2387-2400
[14] Lim R, Khoi L. Complex symmetric weighted composition operators on $H_\gamma (\mathbb{D})$. J Math Anal Appl, 2018, 464: 101-118
[15] Ri M H, Farwig R.Maximal $L^1$-regularity of generators for bounded analytic semigroups in Banach spaces. Acta Math Sci, 2022, 42B: 1261-1272
[16] Siskakis A G. Weighted composition semigroups on Hardy spaces. Linear Algebra Appl, 1986, 84: 359-371
[17] Siskakis A G. Semigroups of composition operators in Bergman spaces. Bull Austral Math Soc, 1987, 35: 397-406
[18] Siskakis A G. Semigroups of composition operators on the Dirichlet space. Results Math, 1996, 30: 165-173
[19] Wu F L. Weighted composition semigroups on some Banach spaces. Complex Anal Oper Theory, 2021, 15: 1-14
[20] Zhao R, Zhu K. Theory of Bergman spaces in the unit ball of $\mathbb{C}^n$. Mem Soc Math Fr, 2008, 115: 1-103
