Hardy spaces on a finite bordered Riemann surface, multivariable operator model theory and Fourier analysis along a unimodular curve

Abstract

In this paper we continue the study of (in general) indefinite Hardy spaces on a finite bordered Riemann surface as reported in [5]. To put the ideas in context, we begin with the classical case, where the bordered Riemann surface is the closed unit disk $\bar{\mathbb{D}}$. We let $\mathbb{T}$ denote the unit circle, ε_* denote a Hilbert space, and let ${{L}^{2}}(\mathbb{T},{{\mathcal{E}}_{*}})$ be the standard Lebesgue space of ε _*-valued, measurable functions f defined on $\mathbb{T}$ with $\parallel f\parallel _{2}^{2} = \tfrac{1}{{2\pi }}{{\smallint }_{\mathbb{T}}}\parallel f(z){{\parallel }^{2}}|dz| < \infty$ Then an arbitrary element f of ${{L}^{2}}(\mathbb{T},{{\mathcal{E}}_{*}})$ alternatively can be presented in terms of a Fourier representation $f(z) \sim \sum _{{n = - \infty }}^{\infty }{{f}_{n}}{{z}^{n}}$ with Fourier coefficients f _n (n =…, -1, 0, 1,…) taking values in ε _* and with $\parallel f\parallel _{2}^{2} = \sum _{{n = - \infty }}^{\infty }\parallel f{{\parallel }^{2}}$ The Hardy space ${{H}^{2}}(\mathbb{D},{{\mathcal{E}}_{*}})$ consists of ε _*.-valued analytic functions on the unit disk $\mathbb{D}$ with $\parallel f(z){{\parallel }^{2}}$ possessing a harmonic majorant, and can be identified as the subspace of functions f in ${{L}^{2}}(\mathbb{T},{{\mathcal{E}}_{*}})$ with Fourier representation of the form $f(z) \sim \sum _{{n = 0}}^{\infty }{{f}_{n}}{{z}^{n}}$ The orthogonal complement ${{H}^{2}}{{(\mathcal{D},{{\mathcal{E}}_{*}})}^{ \bot }}$ in ${{L}^{2}}(\mathbb{T},{{\mathcal{E}}_{*}})$ consists of ${{L}^{2}}(\mathbb{T},{{\mathcal{E}}_{*}})$ -functions f with Fourier series of the form $f(z) \sim \sum _{{n = - \infty }}^{{ - 1}}{{f}_{n}}{{z}^{n}}$ and can be identified with the Hardy space $H_{0}^{2}({{\mathbb{D}}_{e}},{{\mathcal{E}}_{*}})$ consisting of functions analytic on the complement ${{\mathbb{D}}_{e}}$ of the closed unit disk in the extended complex plane ${{\mathbb{C}}_{\infty }}$ which vanish at ∞. The operator M _z of multiplication by the coordinate function z on ${{L}^{2}}\left( {\mathbb{T},{{\varepsilon }_{*}}} \right)$ is unitary (specifically, the bilateral shift operator of multiplicity equal to dim ε _*), and its restriction $V{\varepsilon _*}: = {M_z}{|_{{H^2}\left( {\mathbb{D},{\varepsilon _*}} \right)}}:{H^2}\left( {\mathbb{D},{\varepsilon _*}} \right) \to {H^2}\left( {\mathbb{D},{\varepsilon _*}} \right)$ is an isometry (the unilateral shift operator of multiplicity equal to dim ε _*).