Skip to main content Accessibility help
×
Home

THE ESSENTIAL NORMS OF COMPOSITION OPERATORS ON WEIGHTED DIRICHLET SPACES

  • YUFEI LI (a1), YUFENG LU (a2) and TAO YU (a3)

Abstract

Let $\unicode[STIX]{x1D711}$ be an analytic self-map of the unit disc. If $\unicode[STIX]{x1D711}$ is analytic in a neighbourhood of the closed unit disc, we give a precise formula for the essential norm of the composition operator $C_{\unicode[STIX]{x1D711}}$ on the weighted Dirichlet spaces ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}$ for $\unicode[STIX]{x1D6FC}>0$ . We also show that, for a univalent analytic self-map $\unicode[STIX]{x1D711}$ of $\mathbb{D}$ , if $\unicode[STIX]{x1D711}$ has an angular derivative at some point of $\unicode[STIX]{x2202}\mathbb{D}$ , then the essential norm of $C_{\unicode[STIX]{x1D711}}$ on the Dirichlet space is equal to one.

Copyright

Corresponding author

Footnotes

Hide All

This research is supported by NSFC grant no. 11671065. The third author is supported by the NSFC grant nos. 11271332 and 11431011.

Footnotes

References

Hide All
[1] Carswell, B. and Hammond, C., ‘Composition operators with maximal norm on weighted Bergman spaces’, Proc. Amer. Math. Soc. 134 (2006), 25992605.
[2] Cima, J. A. and Matheson, A. L., ‘Essential norms of composition operators and Aleksandrov measures’, Pacific J. Math. 179 (1997), 5964.
[3] Cowen, C. C., ‘Composition operators on H 2 ’, J. Operator Theory 9 (1983), 77106.
[4] Cowen, C. C. and MacCluer, B. D., Composition Operators on Spaces of Analytic Functions (CRC Press, Boca Raton, 1995).
[5] Garnett, J. B., Bounded Analytic Functions (Springer, New York, 2007).
[6] Hammond, C., ‘The norm of a composition operator with linear symbol acting on the Dirichlet space’, J. Math. Anal. Appl. 303 (2005), 499508.
[7] MacCluer, B. D. and Shapiro, J. H., ‘Angular derivative and compact composition operators on the Hardy and Bergman spaces’, Canad. J. Math. 38 (1986), 878906.
[8] Poggi-Corradini, P., The Essential Norm of Composition Operators Revisited, Contemporary Mathematics, 213 (American Mathematical Society, Providence, RI, 1998), 167173.
[9] Shapiro, J. H., ‘The essential norm of a composition operator’, Ann. of Math. (2) 125(2) (1987), 375404.
[10] Shapiro, J. H., Composition Operators and Classical Function Theory (Springer, New York, 1993).
[11] Shimorin, S., ‘Factorization of analytic functions in weighted Bergman spaces’, St. Petersburg Math. J. 5 (1994), 10051022.
[12] Shimorin, S., ‘On a family of conformally invariant operators’, St. Petersburg Math. J. 7 (1996), 287306.
[13] Shimorin, S., ‘The green function for the weighted biharmonic operator 𝛥(1 -|z|2)-𝛼𝛥, and factorization of analytic functions’, J. Math. Sci. 87 (1997), 39123924.
[14] Zhu, K., Operator Theory in Function Spaces, 2nd edn (American Mathematical Society, Providence, RI, 2007).
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed