Some thoughts on composition operators on subspaces of the Hardy space

  • Jatin Anand
  • Tirthankar Bhattacharyya
  • Sachi SrivastavaEmail author


We discuss composition operators on certain subspaces of the Hardy space. The family of subspaces that we deal with are called \(H^2_{\alpha , \beta }\) which have garnered a lot of attention recently for results related to interpolation. We use them effectively here to study composition operators. Three aspects are discussed. The first is invariance. We examine when \(H^2_{\alpha , \beta }\) or \( J H^2_{\alpha , \beta }\) where J is an inner function are left invariant by composition operators. Secondly, we show that for detecting whether a function \(\varphi \) is inner or not, the composition operator with the symbol \(\varphi \) can be used efficiently on certain subspaces. Thirdly, we discover a criterion for detecting invertibility in the footsteps of the classical result of Schwartz.


Composition operators Invariant subspaces Inner functions 

Mathematics Subject Classification

47B33 47A15 47B38 



The second named author acknowledges the support by the University Grants Commission, India through its Centre for Advanced Studies.


  1. 1.
    Cowen, C.C., MacCluer, B.D.: Composition Operators on Spaces of Analytic Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)zbMATHGoogle Scholar
  2. 2.
    Cowen, C.C., Wahl, R.: Shift-invariant subspaces invariant for composition operators on the Hardy–Hilbert space. Proc. Am. Math. Soc. 142(12), 4143–4154 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Davidson, K.R., Paulsen, V.I., Raghupati, M., Singh, D.: A constrained Nevanlinna–Pick interpolation problem. Indiana Univ. Math. J. 58(2), 709–732 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Jones, M.M.: Shift invariant subspaces of composition operators on \(H^p\). Arch. Math. (Basel) 84(3), 258–267 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Mahvidi, A.: Invariant subspaces of composition operators. J. Oper. Theory 46(3, suppl.), 453–476 (2001)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Martínez-Avendaño, R.A., Rosenthal, P.: An Introduction to Operators on the Hardy–Hilbert Space. Graduate Texts in Mathematics, vol. 237. Springer, New York (2007)zbMATHGoogle Scholar
  7. 7.
    Matache, V.: Invariant subspaces of composition operators. J. Oper. Theory 73(1), 243–264 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Nordgren, E., Rosenthal, P., Wintrobe, F.S.: Invertible composition operators on \(H^p\). J. Funct. Anal. 73(2), 324–344 (1987)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Schwartz, H.J.: Composition operators on \(H^p\). Thesis, University of Toledo (1969)Google Scholar
  10. 10.
    Shapiro, J.H.: What do composition operators know about inner functions? Monatsh. Math. 130(1), 57–70 (2000)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DelhiDelhiIndia
  2. 2.Department of MathematicsIndian Institute of ScienceBengaluruIndia
  3. 3.Department of MathematicsUniversity of DelhiNew DelhiIndia

Personalised recommendations