The norm of a composition operator with linear symbol acting on the Dirichlet space. (English) Zbl 1061.47023

Summary: We obtain a representation for the norm of a composition operator on the Dirichlet space induced by a map of the form \(\varphi(z)=az+b\). We compare this result to an upper bound for \(\| C_{\varphi}\|\) that is valid whenever \(\varphi\) is univalent. Our work relies heavily on an adjoint formula recently discovered by E. A. Gallardo–Gutiérrez and A. Montes–Rodríguez [Math. Ann. 327, 117–134 (2003; Zbl 1048.47016)].


47B33 Linear composition operators
30H05 Spaces of bounded analytic functions of one complex variable
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
31C25 Dirichlet forms


Zbl 1048.47016
Full Text: DOI Link


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