Continuity of the Norm of a Composition Operator

By David Pokorny and Jonathan E. Shapiro

This paper appears in the journal Integral Equations and Operator Theory, Volume 45, Issue 3, March 2003, pp 351-358.


We explore the continuity of the map which, given an analytic self-map of the disk, takes as its value the norm of the associated composition operator on the Hardy space H2. 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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