### Abstract

In this paper we investigate fundamental properties of state-space realizations for inner functions. We derive necessary and sufficient conditions for the inner function to have a realization such that the associated $C_0$-semigroup is exponentially stable. Furthermore, we give necessary and sufficient conditions on the inner function such that the $C_0$-semigroup is a group. Combining these results, we have that the $C_0$-semigroup is an exponentially stable $C_0$-group if and only if the inner function is the product of a constant of modulus one and a Blaschke product for which the zeros satisfy the Carleson-Newman condition and the zeros lie in a vertical strip bounded away from the imaginary axis.

Original language | English |
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Pages (from-to) | 356-379 |

Number of pages | 24 |

Journal | Mathematics of control, signals and systems |

Volume | 15 |

Issue number | 4 |

DOIs | |

Publication status | Published - Nov 2002 |

### Keywords

- MSC-93B15
- Infinite-dimensional systems
- Exponential stability
- Semigroups of operators
- Realization theory
- Inner functions

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## Cite this

Jacob, B., & Zwart, H. (2002). Properties of the realization of inner functions.

*Mathematics of control, signals and systems*,*15*(4), 356-379. https://doi.org/10.1007/s004980200015