Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access March 11, 2021

The Numerical Range of C*ψ Cφ and Cφ C*ψ

  • John Clifford EMAIL logo , Michael Dabkowski and Alan Wiggins
From the journal Concrete Operators


In this paper we investigate the numerical range of C*m Cn and Cn C*m on the Hardy space where φ is an inner function fixing the origin and a and b are points in the open unit disc. In the case when |a| = |b| = 1 we characterize the numerical range of these operators by constructing lacunary polynomials of unit norm whose image under the quadratic form incrementally foliate the numerical range. In the case when a and b are small we show numerical range of both operators is equal to the numerical range of the operator restricted to a 3-dimensional subspace.

MSC 2010: 47B33; 47A12; 15A60


[1] P. Bourdon, J. Shapiro, The numerical range of a composition operators, J. Math. Anal. Appl, 251 (2000), pp. 839-854.Search in Google Scholar

[2] P. Bourdon, J. Shapiro, When is zero in the numerical range of a composition operator?, Integr. Equ. Oper. Theory, 44 (2002), pp. 410-444.Search in Google Scholar

[3] J. H. Clifford, T. Le, A. Wiggins, Invertible composition operators; the product of a composition operator with the adjoint of a composition operator, Complex Anal. Oper. Theory 8 (2014), no. 8, 1699–1706.Search in Google Scholar

[4] C.C. Cowen, Linear fractional composition operators on H2(𝔻), Integral Eqns. Op. Th. 11, (1988), 151-160.10.1007/BF01272115Search in Google Scholar

[5] C.C. Cowen and B.D. MacCluer, Composition operators on spaces of analytic functions, CRC Press, (1995).Search in Google Scholar

[6] K. Gustafson and D. Rao, Numerical Range, The Field of Values of Linear Operators and Matrices, Springer-Verlag, (1997)10.1007/978-1-4613-8498-4_1Search in Google Scholar

[7] U. Haagerup and P. dela Harpe, The numerical radius of a nilpotent operator on a Hilbert space, Proc. Amer. Math. Soc. 115(1992), 371–379.10.1090/S0002-9939-1992-1072339-6Search in Google Scholar

[8] J. E. Littlewood, On inequalities in the theory of functions, Proc. London Math. Soc. 23 (1925).10.1112/plms/s2-23.1.481Search in Google Scholar

[9] V. Matache, Numerical Ranges of composition operators, Linear Algebra Appl. 331 (2001), pp. 61-74Search in Google Scholar

[10] R.A. Martinez and P. Rosenthal, An Introduction to Operators on Hardy-Hilbert Space, Springer-Verlag, 2007.Search in Google Scholar

[11] E. A. Nordgren, Composition operators, Canad. J. Math. 20(1968), 442-449.10.4153/CJM-1968-040-4Search in Google Scholar

[12] J. H. Shapiro, Composition operators and classical function theory, Springer-Verlag, 1993.10.1007/978-1-4612-0887-7Search in Google Scholar

[13] A. Shields, Weighted Shift Operators and Analytic Function Theory, Topics in Operator Theory, 13, 1974, pp. 49-128.10.1090/surv/013/02Search in Google Scholar

[14] B. Simon, A Comprehensive Course in Analysis, Part 4: Operator Theory, American Mathematical Society, Providence, RI, 2015.Search in Google Scholar

[15] Q. Stout, The numerical range of a shift, Proc. Amer. Math. Soc. 88 (1983), pp. 495-502.Search in Google Scholar

Received: 2020-09-16
Accepted: 2021-01-05
Published Online: 2021-03-11

© 2020 John Clifford et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 1.4.2023 from
Scroll to top button