It is well known that Hardy and weighted Bergman spaces of the upper half-plane do not support compact composition operators (see [M99] and [SS03]). In this paper, we prove that unlike Hardy and Bergman spaces, the Dirichlet space of the upper half-plane does support compact composition operators. Furthermore, bounded analytic symbols, which in the case of Hardy and weighted Bergman spaces of the upper half-plane do not even induce bounded composition operators, can induce compact composition operators on the Dirichlet space of the upper half-plane.

The first author was supported by the research grant 02011/30/2017/R&D II/12565 of NBHM(DAE) (India).