Complex symmetry and cyclicity of composition operators on 𝐻²(ℂ₊)

Noor, S.; Severiano, Osmar

doi:10.1090/proc/14918

Complex symmetry and cyclicity of composition operators on $H^2(\mathbb {C}_+)$
HTML articles powered by AMS MathViewer

by S. Waleed Noor and Osmar R. Severiano PDF

Proc. Amer. Math. Soc. 148 (2020), 2469-2476 Request permission

Abstract:

In this article, we completely characterize the complex symmetry, cyclicity, and hypercyclicity of composition operators $C_\phi f=f\circ \phi$ induced by affine self-maps $\phi$ of the right half-plane $\mathbb {C}_+$ on the Hardy-Hilbert space $H^2(\mathbb {C}_+)$. The interplay between complex symmetry and cyclicity plays a key role in the analysis. We also provide new proofs for the normal, self-adjoint, and unitary cases and for an adjoint formula discovered by Gallardo-Gutiérrez and Montes-Rodríguez.

References

Frédéric Bayart and Étienne Matheron, Dynamics of linear operators, Cambridge Tracts in Mathematics, vol. 179, Cambridge University Press, Cambridge, 2009. MR 2533318, DOI 10.1017/CBO9780511581113
Paul S. Bourdon and Joel H. Shapiro, Cyclic phenomena for composition operators, Mem. Amer. Math. Soc. 125 (1997), no. 596, x+105. MR 1396955, DOI 10.1090/memo/0596
Sam Elliott and Michael T. Jury, Composition operators on Hardy spaces of a half-plane, Bull. Lond. Math. Soc. 44 (2012), no. 3, 489–495. MR 2966995, DOI 10.1112/blms/bdr110
Eva A. Gallardo-Gutiérrez and Alfonso Montes-Rodríguez, Adjoints of linear fractional composition operators on the Dirichlet space, Math. Ann. 327 (2003), no. 1, 117–134. MR 2005124, DOI 10.1007/s00208-003-0442-9
Eva A. Gallardo-Gutiérrez and Alfonso Montes-Rodríguez, The role of the spectrum in the cyclic behavior of composition operators, Mem. Amer. Math. Soc. 167 (2004), no. 791, x+81. MR 2023381, DOI 10.1090/memo/0791
Stephan Ramon Garcia and Christopher Hammond, Which weighted composition operators are complex symmetric?, Concrete operators, spectral theory, operators in harmonic analysis and approximation, Oper. Theory Adv. Appl., vol. 236, Birkhäuser/Springer, Basel, 2014, pp. 171–179. MR 3203059, DOI 10.1007/978-3-0348-0648-0_{1}0
Stephan Ramon Garcia and Mihai Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 (2006), no. 3, 1285–1315. MR 2187654, DOI 10.1090/S0002-9947-05-03742-6
Stephan Ramon Garcia and Mihai Putinar, Complex symmetric operators and applications. II, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3913–3931. MR 2302518, DOI 10.1090/S0002-9947-07-04213-4
Stephan Ramon Garcia and Warren R. Wogen, Complex symmetric partial isometries, J. Funct. Anal. 257 (2009), no. 4, 1251–1260. MR 2535469, DOI 10.1016/j.jfa.2009.04.005
Stephan Ramon Garcia and Warren R. Wogen, Some new classes of complex symmetric operators, Trans. Amer. Math. Soc. 362 (2010), no. 11, 6065–6077. MR 2661508, DOI 10.1090/S0002-9947-2010-05068-8
John B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007. MR 2261424
Karl-G. Grosse-Erdmann and Alfredo Peris Manguillot, Linear chaos, Universitext, Springer, London, 2011. MR 2919812, DOI 10.1007/978-1-4471-2170-1
Valentin Matache, Composition operators on Hardy spaces of a half-plane, Proc. Amer. Math. Soc. 127 (1999), no. 5, 1483–1491. MR 1625773, DOI 10.1090/S0002-9939-99-05060-1
Valentin Matache, Weighted composition operators on $H^2$ and applications, Complex Anal. Oper. Theory 2 (2008), no. 1, 169–197. MR 2390678, DOI 10.1007/s11785-007-0025-y
Valentin Matache, Invertible and normal composition operators on the Hilbert Hardy space of a half-plane, Concr. Oper. 3 (2016), no. 1, 77–84. MR 3505680, DOI 10.1515/conop-2016-0009
Sivaram K. Narayan, Daniel Sievewright, and Derek Thompson, Complex symmetric composition operators on $H^2$, J. Math. Anal. Appl. 443 (2016), no. 1, 625–630. MR 3508506, DOI 10.1016/j.jmaa.2016.05.046
Tami Worner, Commutants of certain composition operators, Acta Sci. Math. (Szeged) 68 (2002), no. 1-2, 413–432. MR 1916588