## A State Space Approach to Canonical Factorization with ApplicationsThe present book deals with canonical factorization problems for di?erent classes of matrix and operator functions. Such problems appear in various areas of ma- ematics and its applications. The functions we consider havein common that they appear in the state space form or can be represented in such a form. The main results are all expressed in terms of the matrices or operators appearing in the state space representation. This includes necessary and su?cient conditions for canonical factorizations to exist and explicit formulas for the corresponding f- tors. Also, in the applications the entries in the state space representation play a crucial role. Thetheorydevelopedinthebookisbasedonageometricapproachwhichhas its origins in di?erent ?elds. One of the initial steps can be found in mathematical systems theory and electrical network theory, where a cascade decomposition of an input-output system or a network is related to a factorization of the associated transfer function. Canonical factorization has a long and interesting history which starts in the theory of convolution equations. Solving Wiener-Hopf integral equations is closely related to canonical factorization. The problem of canonical factorization also appears in other branches of applied analysis and in mathematical systems theory, in H -control theory in particular. |

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### Contents

1 | |

7 | |

8 | |

Chapter 2 The state space method and factorization | 19 |

Part II Convolution equations with rational matrix symbols | 34 |

Chapter 3 Explicit solutions using realizations | 37 |

Chapter 4 Factorization of nonproper rational matrix functions | 57 |

Part III Equations with nonrational symbols | 75 |

Part V Riccati equations and factorization | 217 |

Chapter 12 Canonical factorization and Riccati equations | 218 |

Chapter 13The symmetric algebraic Riccatiequation | 233 |

Chapter 14 Jspectral factorization | 248 |

Part VI Factorizations and symmetries | 289 |

Chapter 15 Factorization of positive realrational matrix functions | 291 |

Chapter 16 Contractive rational matrix functions | 301 |

Chapter 17 Junitary rational matrix functions | 313 |

Chapter 5 Factorization of matrix functions analytic in a strip | 76 |

Chapter 6 Convolution equations and the transport equation | 115 |

Chapter 7 WienerHopf factorization and factorization indices | 143 |

Part IV Factorization of selfadjoint rational matrix functions | 168 |

Chapter 8 Preliminaries concerning minimal factorization | 171 |

Chapter 9 Factorization of positive definite rational matrix functions | 180 |

Chapter 10 Pseudospectral factorizations of selfadjoint rational matrix functions | 197 |

Chapter 11 Review of the theory of matrices in indefinite inner product spaces | 211 |

Part VII Applications of Jspectral
factorizations | 346 |

Chapter 18 Application to the rational Nehari problem | 349 |

Chapter 19 Review of some control theory for linear systems | 371 |

Chapter 20 Hinfinity control applications | 379 |

405 | |

List of symbols | 415 |

419 | |

### Other editions - View all

A State Space Approach to Canonical Factorization with Applications Harm Bart,Israel Gohberg,Marinus Kaashoek,André C.M. Ran No preview available - 2010 |

### Common terms and phrases

A-invariant admits a left admits a right algebraic Riccati equation analytic assume biproper bounded linear operator C(XI chapter corresponding defined eigenvalues equivalent factorization with respect follows formulas given Hence Hermitian matrix Hermitian solution identity imaginary axis implies infinity included inner product invertible matrix J-unitary rational KerP left canonical factorization left half plane left J-spectral factorization Lemma lower half plane minimal factorization minimal realization Nehari problem nonnegative obtained open left half open right half open unit disc pair poles positive definite projection proof of Theorem proper rational Proposition prove pseudo-canonical factorization pseudo-spectral factorization rational matrix functions real line realization triple result Riccati equation Riemann sphere right canonical factorization right half plane right J-spectral factorization satisfied Section selfadjoint signature matrix space spectral subspace theory Toeplitz operator unit circle upper half plane Wiener-Hopf zero