Domination of Berezin Transform

Abstract

Let 𝔻 be the open unit disk in the complex plane ℂ and \(dA(z) = \frac {1}{\pi }dx dy\), the normalized area measure on 𝔻. Let \({L_{a}^{2}}(\mathbb {D}, dA)\) be the Bergman space consisting of analytic functions on 𝔻 that are also in \(L^{2}(\mathbb {D}, dA)\). Let \(\mathcal {L}({L_{a}^{2}}(\mathbb {D}))\) be the set of all bounded linear operators from the Hilbert space \({L_{a}^{2}}(\mathbb {D})\) into itself. For \(T\in \mathcal {L}({L_{a}^{2}}(\mathbb {D}))\), let \(\widetilde {T}\) denote the Berezin transform of T. In this paper, we find conditions on positive operators \(S, T\in \mathcal {L}({L_{a}^{2}}(\mathbb {D}))\) such that \(\widetilde {S}(z)\geq \widetilde {T}(z)\) for all z ∈ 𝔻.