## The Hardy Space of a Slit DomainIf H is a Hilbert space and T : H ? H is a continous linear operator, a natural question to ask is: What are the closed subspaces M of H for which T M ? M? Of course the famous invariant subspace problem asks whether or not T has any non-trivial invariant subspaces. This monograph is part of a long line of study of the invariant subspaces of the operator T = M (multiplication by the independent variable z, i. e. , M f = zf )on a z z Hilbert space of analytic functions on a bounded domain G in C. The characterization of these M -invariant subspaces is particularly interesting since it entails both the properties z of the functions inside the domain G, their zero sets for example, as well as the behavior of the functions near the boundary of G. The operator M is not only interesting in its z own right but often serves as a model operator for certain classes of linear operators. By this we mean that given an operator T on H with certain properties (certain subnormal operators or two-isometric operators with the right spectral properties, etc. ), there is a Hilbert space of analytic functions on a domain G for which T is unitarity equivalent to M . |

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### Contents

Introduction | 1 |

12 Invariant subspaces of the slit disk | 2 |

13 Nearly invariant subspaces | 5 |

14 Cyclic invariant subspaces | 6 |

15 Essential spectrum | 7 |

Preliminaries | 9 |

22 Harmonic measure | 12 |

23 Slit domains | 15 |

52 de Branges spaces and nearly invariant subspaces | 60 |

Invariant subspaces of the slit disk | 64 |

62 Second description of the invariant subspaces | 68 |

Cyclic invariant subspaces | 79 |

72 Cyclic subspaces | 80 |

73 Polynomial approximation | 82 |

The essential spectrum | 84 |

Other applications | 93 |

24 More about the Hardy space | 20 |

Nearly invariant subspaces | 24 |

32 Normalized reproducing kernels | 26 |

33 The operator J | 34 |

34 The Wold decomposition | 37 |

35 Proof of the main theorem | 42 |

36 Uniqueness of the parameters | 46 |

Nearly invariant and the backward shift | 47 |

42 A new description of nearly invariant subspaces | 48 |

Nearly invariant and de Branges spaces | 59 |

92 The parameters | 95 |

Domains with several slits | 97 |

102 Some technical lemmas | 99 |

103 A localization of Yakubovich | 101 |

104 Finally the proof | 111 |

Final thoughts | 113 |

Appendix | 115 |

117 | |

122 | |

### Other editions - View all

The Hardy Space of a Slit Domain Alexandru Aleman,Nathan S. Feldman,William T. Ross No preview available - 2010 |

The Hardy Space of a Slit Domain Alexandru Aleman,Nathan S. Feldman,William T. Ross No preview available - 2009 |

### Common terms and phrases

analytic continuation analytic functions analytic polynomials Beurling’s theorem boundary function boundary values Cauchy integral formula Cauchy transform closed graph theorem common zero computation conformal map converges Corollary Cyclic invariant subspaces D-outer function defined definition domain G equal essential spectrum extremal function f,ge fact Fatou's jump theorem fe H*(G function F Furthermore Hardy space harmonic majorant harmonic measure Hilbert space identity inequality inner factor inner function inner product isometry Jordan domain least harmonic majorant Lebesgue measure limits almost everywhere linear manifold main theorem measurable function measurable set Mz-invariant subspaces nearly invariant subspace non-zero normalized reproducing kernel norming point notation notice operator piecewise analytic pointwise positive measure proof of Theorem prove pseudocontinuations Recall reproducing kernel function S-invariant satisfies says sequence simply connected slit disk slit domain space of analytic subspace of H*(G Suppose unique Wold decomposition Y-inner divisor