## 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)].

### MSC:

 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

### Keywords:

composition operator; Dirichlet space

Zbl 1048.47016
Full Text:

### References:

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