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Continuity of the Norm of a Composition Operator

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Abstract.

We explore the continuity of the map which, given an analytic selfmap of the disk, takes as its value the norm of the associated composition operator on the Hardy space \( H^{2} \). We also examine the continuity the functions which assign to a self-map of the disk the Hilbert-Schmidt norm or the essential norm of the associated composition operator and show these to be discontinuous. Additionally, we characterize when the norm of a composition operator is minimal.

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Authors and Affiliations

  1. Mathematics Department, University of California, Berkeley, CA 94720, USA. E-mail: davebrok@soda.csua.berkeley.edu ¶URL: www.csua.berkeley.edu/~davebrok, , , , , , US

    David B. Pokorny

  2. Mathematics Department, California Polytechnic State University, San Luis Obispo, CA 93407, USA. E-mail: jshapiro@calpoly.edu ¶URL: http://www.calpoly.edu/~jshapiro, , , , , , US

    Jonathan E. Shapiro

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  1. David B. Pokorny
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  2. Jonathan E. Shapiro
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Submitted: January 3, 2002¶ Revised: March 1, 2002.

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Pokorny, D., Shapiro, J. Continuity of the Norm of a Composition Operator. Integr. equ. oper. theory 45, 351–358 (2003). https://doi.org/10.1007/s000200300010

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  • DOI: https://doi.org/10.1007/s000200300010

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