A note on composition operators on the disc and bidisc

Beslikas, Athanasios 

Canadian mathematical bulletin, Tome 68 (2025) no. 4, pp. 1163-1176

Résumé

In this note, we give a new necessary condition for the boundedness of the composition operator on the Dirichlet-type space on the disc, via a two dimensional change of variables formula. With the same formula, we characterize the bounded composition operators on the anisotropic Dirichlet-type spaces $\mathfrak {D}_{\vec {a}}(\mathbb {D}^2)$ induced by holomorphic self maps of the bidisc $\mathbb {D}^2$ of the form $\Phi (z_1,z_2)=(\phi _1(z_1),\phi _2(z_2))$. We also consider the problem of boundedness of composition operators $C_{\Phi }:\mathfrak {D}(\mathbb {D}^2)\to A^2(\mathbb {D}^2)$ for general self maps of the bidisc, applying some recent results about Carleson measures on the Dirichlet space of the bidisc.

Détail

DOI : 10.4153/S0008439525000359

Mots-clés : Dirichlet-type spaces, composition operators