BEREZIN TRANSFORM OF INVERTIBLE POSITIVE OPERATORS

Abstract

In this paper we introduce a class A ⊂ L∞(D) such that if φ ∈ Aand satisfies certain positive-definite condition, then there exists aψ ∈ A such that φ(z) ≤ αeψ(z), for some constant α > 0. Further,if φ(z) = hAkz, kzi, for some bounded positive, invertible operator Afrom the Bergman space L2a(D) into itself then ψ(z) = h(log A)kz, kzi.Here kz, z ∈ D are the normalized reproducing kernel of L2a(D). Ap-plications of these results are also discusseonly a non-standard growth condition. We show that our problem admits at least one weak solution. In order to do this, the main tool is the Berkovits degree theory for abstract Hammerstein type mappings.