On a class of unitary operators on the Bergman space of the right half plane

In this paper, we introduce a class of unitary operators defined on the Bergman space La(C+) of the right half plane C+ and study certain algebraic properties of these operators. Using these results, we then show that a bounded linear operator S from La(C+) into itself commutes with all the weighted composition operators Wa, a ∈ D if and only if S̃(w) = ⟨Sbw, bw⟩, w ∈ C+ satisfies a certain averaging condition. Here for a = c + id ∈ D, f ∈ La(C+),Waf = (f ◦ ta) M ′ M′◦ta ,Ms = 1−s 1+s , ta(s) = −ids+(1−c) (1+c)s+id , and bw(s) = 1 √ π 1+w 1+w 2Rew (s+w)2 , w = Ma, s ∈ C+. Some applications of these results are also discussed.