Path components of composition operators over the half-plane

The characterization of path components in the space of composition operators acting in various settings has been a long-standing open problem. Recently Dai has obtained a characterization of when two composition operators acting on the weighted Hilbert-Bergman space on the unit disk are linearly connected, i.e., they are joined by a continuous “line segment” of composition operators induced by convex combinations of the maps inducing the two given composition operators. In this paper we consider composition operators acting on the weighted Bergman spaces over the half-plane. Since not all composition operators are bounded in this setting, we introduce a metric induced by the operator norm and study when (possibly unbounded) composition operators are linearly connected in the resulting metric space. We obtain necessary conditions that under a natural additional assumption are also sufficient. We also study the problem of when a composition operator is isolated. Complete results are obtained for composition operators induced by linear fractional self-maps of the half-plane. We show that the only such composition operators that are isolated are those induced by automorphisms of the half-plane. We also characterize when composition operators induced by linear fractional self-maps belong to the same path component. The characterization demonstrates that composition operators in the same path component may have inducing maps with different behavior at infinity. In contrast, when the setting is the disk the corresponding boundary behavior of the inducing maps must match.