### Abstract

In this paper there is introduced and studied the following characteristic of a linear operator A acting on a Banach space Χ:[Figure not available: see fulltext.], where Cyc A=R:R is a subspace of Χ, dim R<+∞. Spqn (A^{n}R:n≥0)=χ. Always disc A ≥μ_{A}=(the multiplicity of the spectrum of the operator[Figure not available: see fulltext.] (dim R:R∈Cyc A), where (by definition) in each A-cyclic subspace there is contained a cyclic subspace of dimension ≤ disc A. For a linear dynamical system x(t)=Ax(t)+Bu,(t) which is controllable, the characteristic disc A of the evolution operator A shows how much the control space can be diminished without losing controllability. In this paper there are established some general properties of disc (for example, conditions are given under which disc(A⊕B))=max(discA, disc B); disc is computed for the following operators: S (S is the shift in the Hardy space H^{2}); disc S=2, (but μ_{S}=i); disc S_{n}^{*}=n (but μ=1), where S_{n}=S⊕. ⊕S; disc S=2, (but μ_{S}=1), where S is the bilateral shift. It is proved that for a normal operator N with simple spectrum, disc N=μ_{N}=1 {mapping} (the operator N is reductive). There are other results also, and also a list of unsolved problems.

Original language | English |
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Pages (from-to) | 1719-1738 |

Number of pages | 20 |

Journal | Journal of Soviet Mathematics |

Volume | 22 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1 Aug 1983 |

### Scopus subject areas

- Statistics and Probability
- Mathematics(all)
- Applied Mathematics

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## Cite this

*Journal of Soviet Mathematics*,

*22*(6), 1719-1738. https://doi.org/10.1007/BF01882576