Spectral properties of holomorphic automorphism with fixed point

Abstract

The Hilbert space methods in the theory of biholomorphic mappings were applied and developed by S. Bergman [1, 2]. In this approach the central role is played by the Hilbert spaceL2H(D)consisting of all functions which are square integrable and holomorphic in a domainD ⊂ ℂN. A biholomorphic mapping φ:D ⃗ Ginduces the unitary mappingUφ:L2H(G)⃗L2H(D)defined by the formulaHere ∂φ/∂z denotes the complex Jacobian of φ. The mappingUϕis useful, since it permits to replace a problem forDby a problem for its biholomorphic imageG(see for example [11], [13]). When ϕ is an automorphism ofDwe obtain a unitary operatorUϕonL2H(D).