| Beurling's theorem and invariant subspaces for the shift on Hardy spaces |
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| 引用本文： | QIU ZhiJian School of Economic Mathematics,Southwestern University of Finance and Economics,Chengdu 610074,China. Beurling's theorem and invariant subspaces for the shift on Hardy spaces[J]. 中国科学A辑(英文版), 2008, 51(1): 131-142. DOI: 10.1007/s11425-007-0184-3 |
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| 作者姓名： | QIU ZhiJian School of Economic Mathematics Southwestern University of Finance and Economics Chengdu 610074 China |
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| 作者单位： | QIU ZhiJian School of Economic Mathematics,Southwestern University of Finance and Economics,Chengdu 610074,China |
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| 基金项目： | This work was supported By SWUFE's Key Subjects Construction Items Funds of 211 Project of the 11th Five-Year Plan |
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| 摘 要： | Let G be a bounded open subset in the complex plane and let H~2(G) denote the Hardy space on G. We call a bounded simply connected domain W perfectly connected if the boundary value function of the inverse of the Riemann map from W onto the unit disk D is almost 1-1 with respect to the Lebesgue measure on D and if the Riemann map belongs to the weak-star closure of the polynomials in H~∞(W). Our main theorem states: in order that for each M∈Lat (M_z), there exist u∈H~∞(G) such that M=∨{uH~2(G)}, it is necessary and sufficient that the following hold: (1) each component of G is a perfectly connected domain; (2) the harmonic measures of the components of G are mutually singular; (3) the set of polynomials is weak-star dense in H~∞(G). Moreover, if G satisfies these conditions, then every M∈Lat (M_z) is of the form uH~2(G), where u∈H~∞(G) and the restriction of u to each of the components of G is either an inner function or zero.
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Beurling’s theorem and invariant subspaces for the shift on Hardy spaces |
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| Qiu ZhiJian. Beurling’s theorem and invariant subspaces for the shift on Hardy spaces[J]. Science in China(Mathematics), 2008, 51(1): 131-142. DOI: 10.1007/s11425-007-0184-3 |
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| Authors: | Qiu ZhiJian |
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| Affiliation: | (1) School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 610074, China |
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| Abstract: | Let G be a bounded open subset in the complex plane and let H 2(G) denote the Hardy space on G. We call a bounded simply connected domain W perfectly connected if the boundary value function of the inverse of the Riemann map from W onto the unit disk D is almost 1–1 with respect to the Lebesgue measure on ∂D and if the Riemann map belongs to the weak-star closure of the polynomials in H ∞(W). Our main theorem states: in order that for each M ∈ Lat (M z ), there exist u ∈ H ∞(G) such that M = ∨{uνH 2(G)}, it is necessary and sufficient that the following hold: | (1) | each component of G is a perfectly connected domain | | (2) | the harmonic measures of the components of G are mutually singular | | (3) | the set of polynomials is weak-star dense in H ∞(G). | Moreover, if G satisfies these conditions, then every M ∈ Lat (M z ) is of the form uH 2(G), where u ∈ H ∞(G) and the restriction of u to each of the components of G is either an inner function or zero. This work was supported By SWUFE’s Key Subjects Construction Items Funds of 211 Project of the 11th Five-Year Plan |
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| Keywords: | Hardy space invariant subspace shift operator |
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