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 *).
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Ball, J.A., Vinnikov, V. (2001). Hardy spaces on a finite bordered Riemann surface, multivariable operator model theory and Fourier analysis along a unimodular curve. In: Borichev, A.A., Nikolski, N.K. (eds) Systems, Approximation, Singular Integral Operators, and Related Topics. Operator Theory: Advances and Applications, vol 129. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8362-7_2
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