## Annals of Functional Analysis

### A group structure on $\mathbb{D}$ and its application for composition operators

#### Abstract

We present a group structure on $\mathbb{D}$ via the automorphisms which fix the point $1$. Through the induced group action, each point of $\mathbb{D}$ produces an equivalence class that turns out to be a Blaschke sequence. We show that the corresponding Blaschke products are minimal/atomic solutions of the functional equation $\psi\circ\varphi=\lambda\psi$, where $\lambda$ is a unimodular constant and $\varphi$ is an automorphism of the unit disk. We also characterize all Blaschke products that satisfy this equation, and we study its application in the theory of composition operators on model spaces $K_{\Theta}$.

#### Article information

Source
Ann. Funct. Anal., Volume 7, Number 1 (2016), 76-95.

Dates
Accepted: 23 June 2015
First available in Project Euclid: 6 November 2015

https://projecteuclid.org/euclid.afa/1446819350

Digital Object Identifier
doi:10.1215/20088752-3320401

Mathematical Reviews number (MathSciNet)
MR3449341

Zentralblatt MATH identifier
1343.30047

Subjects
Primary: 30D50
Secondary: 47B33: Composition operators

#### Citation

Fricain, Emmanuel; Karaki, Muath; Mashreghi, Javad. A group structure on $\mathbb{D}$ and its application for composition operators. Ann. Funct. Anal. 7 (2016), no. 1, 76--95. doi:10.1215/20088752-3320401. https://projecteuclid.org/euclid.afa/1446819350

#### References

• [1] A. Beardon, Iteration of Rational Functions, Grad. Texts in Math. 132, Springer, 1991.
• [2] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1949), 239–255.
• [3] J. Cima and W. Ross, The Backward Shift on the Hardy Space, Math. Surv. Monogr. 79, Amer. Math. Soc., Providence, 2000.
• [4] C. Cowen, Iteration and the solution of functional equations for functions analytic in the unit disk, Trans. Amer. Math. Soc. 265 (1981), no. 1, 69–95.
• [5] O. Frostman, Sur les produits de Blaschke, Proc. Roy. Physiog. Soc. Lund 12 (1942), no. 15, 169–182.
• [6] E. Gallardo-Gutirrez, P. Gorkin, and D. Suàrez, Orbits of non-elliptic disc automorphisms on $H^{p}$, J. Math. Anal. Appl. 388 (2012), no. 2, 103–1026.
• [7] G. Königs, Recherches sur les intégrales de certaines équationes functionnelles, Annales Ecole Normale Superior 3 (1884), 3–41.
• [8] Y. Lyubarskii and E. Malinnikova, “Composition operators on model spaces” in Recent Trends in Analysis: Proceedings of the Conference in Honor of Nikolai Nikolski (Bordeaux, 2011), Theta Foundation, Bucharest, 2013, 149–157.
• [9] J. Mashreghi, Representation Theorems in Hardy Spaces, London Math. Soc. Stud. Text Ser. 74, Cambridge Univ. Press, Cambridge, 2009.
• [10] J. Mashreghi and M. Shabankhah, Composition operators on finite rank model subspaces, Glasgow Math. J. 55 (2013), no. 1, 69–83.
• [11] J. Mashreghi and M. Shabankhah, Composition of inner functions, Canad. J. Math. 66 (2014), no. 2, 387–399.
• [12] V. Matache, “The eigenfunctions of a certain composition operator” in Studies on Composition Operators (Laramie, WY, 1996), Contemp. Math. 213, Amer. Math. Soc., Providence, 1998, 121–136.
• [13] T. Needham, Visual Complex Analysis, Oxford Univ. Press, New York, 2002.
• [14] J. Shapiro, Composition Operators and Classical Function Theory, Univeritext: Tracts in Math., Springer, New York, 1993.