Composition of Inner Functions

Mashreghi, J. ; Shabankhah, M.

Canadian journal of mathematics, Tome 66 (2014) no. 2, pp. 387-399

Résumé

We study the image of the model subspace ${{K}_{\theta }}$ under the composition operator ${{C}_{\varphi }}$ , where $\varphi $ and $\theta $ are inner functions, and find the smallest model subspace which contains the linear manifold ${{C}_{\varphi }}{{K}_{\theta }}$ . Then we characterize the case when ${{C}_{\varphi }}$ maps ${{K}_{\theta }}$ into itself. This case leads to the study of the inner functions $\varphi $ and $\psi $ such that the composition $\psi \,\text{o}\,\varphi $ is a divisor of $\psi $ in the family of inner functions.

Détail

DOI : 10.4153/CJM-2013-002-5

Mots-clés : 30D55, 30D05, 47B33, composition operators, inner functions, Blaschke products, model subspaces

Bibliographie

[1] [1] Beurling, A., On two problems concerning linear transformations in Hilbert space. Acta Math. 81(1949), 239–255. Google Scholar

[2] [2] Chen, H. and Gauthier, P., Composition operators of μ-Bloch spaces. Canad. J. Math. 61(2009), 50–75. Google Scholar | DOI

[3] [3] Cowen, C., Iteration and the solution of functional equations for functions analytic in the unit disk. Trans. Amer. Math. Soc. 265(1981), 69–95. Google Scholar | DOI

[4] [4] Cowen, C. and MacCluer, B. D., Composition operators on spaces of analytic functions. Stud. Adv. Math. CRC Press, Boca Raton, FL, 1995. Google Scholar

[5] [5] Douglas, R. G., Shapiro, H. S. and Shields, A. L., Cyclic vectors and invariant subspaces for the backward shift operator. Ann. Inst. Fourier (Grenoble) 20(1970), fasc. 1, 37–76. Google Scholar | DOI

[6] [6] Gallardo-Gutiérrez, Eva A. and González, Maria J., Exceptional sets and Hilbert–Schmidt composition operators. J. Funct. Anal. 199(2003) 287–300. Google Scholar | DOI

[7] [7] Gallardo-Gutiérrez, Eva A., Hausdorff measures, capacities and compact composition operators. Math. Z. 253(2006), 63–74. Google Scholar | DOI

[8] [8] Frostman, O., Sur les produits de Blaschke. (French) Kungl. Fysiografiska Sällskapets i Lund Förhandlingar [Proc. Roy. Physiog. Soc. Lund] 12(1942), 169–182. Google Scholar

[9] [9] MacCluer, B., Compact composition operators in Hp(B). Michigan Math. J. 32(1985), 237–248. Google Scholar | DOI

[10] [10] Mashreghi, J., Representation Theorems in Hardy Spaces. London Math. Soc. Stud. Texts 74, Cambridge University Press, 2009. Google Scholar

[11] [11] Mashreghi, J. and Shabankhah, M., Composition operators on finite rank model subspaces. Glasgow Math. J. 55(2013), 69–83. Google Scholar

[12] [12] Shapiro, J. H., The essential norm of a composition operators. Ann. of Math. 125(1987), 375–404. Google Scholar | DOI

[13] [13] Shapiro, J. H., Composition Operators and Classical Function Theory. Universitext Tracts Math., Springer-Verlag, 1993. Google Scholar

[14] [14] Tjani, M., Compact composition operators on Besov spaces. Trans. Amer. Math. Soc. 355(2003), 4683–4698. Google Scholar | DOI

[15] [15] Zorboska, N., Composition operators on weighted Dirichlet spaces. Proc. Amer. Math. Soc. 126(1998), 2013–2023. Google Scholar | DOI