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Spectral properties of holomorphic automorphism with fixed point

Published online by Cambridge University Press:  18 May 2009

T. Mazur  and
M. Skwarczyński
T. Mazur
Affiliation:
Maciej Skwarczyński, 01698 Warsaw, Smolenskiego 27a m. 14, Poland
M. Skwarczyński
Affiliation:
Maciej Skwarczyński, 01698 Warsaw, Smolenskiego 27a m. 14, Poland
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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 space L2H(D) consisting of all functions which are square integrable and holomorphic in a domain D ⊂ ℂN. A biholomorphic mapping φ:D ⃗ G induces the unitary mapping Uφ:L2H(G) ⃗ L2H(D) defined by the formula

Here ∂φ/∂z denotes the complex Jacobian of φ. The mapping Uϕ is useful, since it permits to replace a problem for D by a problem for its biholomorphic image G (see for example [11], [13]). When ϕ is an automorphism of D we obtain a unitary operator Uϕ on L2H(D).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 28 , Issue 1 , January 1986 , pp. 25 - 30
Copyright
Copyright © Glasgow Mathematical Journal Trust 1986

References

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