×

Harmonic analysis in a mutliply connected domain. II. (Russian) Zbl 0715.47043

Let \({\hat \Omega}\) be a compact Riemannian surface of genus n-1. It can be considered as the “doubling” of two identity copies \(\Omega_+=\Omega_ -=\Omega\) of some n-connected (complex) domain \(\Omega\) with the identified boundary \(\partial \Omega =\cup^{n}_{i=1}\Gamma_ i\) consisting of n piecewise analytic curves \(\Gamma_ 1,...,\Gamma_ n\). The unified complex structure on \(\Omega_+\) and \(\Omega_ -\) are introduced following the principle of symmetry.
Harmonic analysis in the work under reviewing means a decomposition into the sum of Hardy spaces \(H^ 2_{\pm}(\Gamma)\) and the defect space \({\mathcal M}=L^ 2(\Gamma)\odot \{H^ 2_+(\Gamma)+H^ 2_ - (\Gamma)\}\), which can be described by the holomorphic differentials on \({\hat \Omega}\), and the spectral analysis of dynamical operators, following Lax-Phillip theory.
In § 2 the author describes the socalled inner functions which are unimodular on \(\Gamma\) and analytic on \({\bar \Omega}{}_+\), also space \({\mathcal M}\oplus H^ 2_+(\Gamma)\) and construct the Riesz basis for \({\mathcal M}\oplus H^ 2_+(\Gamma)\) and for \(H^ 2_+(\Omega_+)\). In § 3 he shows that the uniform algebra \(B(\Omega_+)\) can be generated by three n-valent inner functions \(\theta_ 0\), \(\theta_ 1\), \(\theta_ 2\) and \(H^ 2_+(\Gamma)=\bigvee^{\infty}_{m,p,q=0}\{\theta^ m_ 0\theta^ p_ 1\theta^ q_ 2\}.\)
He obtain then 2- or 3-parameter unitary groups of translations in \(H^ 2_+(\Gamma)\). He pointed out, in § 5, the subspace, invariant with respect to these groups of translations and finally, in § 6, the spectral analysis of the contractive semigroups, obtained by “cut off” the groups of translations onto the translation-invariant subspace.
The theory generalizes the well known Lax-Phillip theory in the case \(\Gamma ={\mathbb{T}}^ 1\), to the more general case where the continuous spectrum of the dynamical generators have more complicate structure, for example, some zonal structure.
Reviewer: Diep Do Ngoc

MSC:

47N50 Applications of operator theory in the physical sciences
46N50 Applications of functional analysis in quantum physics
47A40 Scattering theory of linear operators
81U40 Inverse scattering problems in quantum theory
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
47A10 Spectrum, resolvent
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces

Citations:

Zbl 0715.47042
PDFBibTeX XMLCite
Full Text: EuDML