## Abstract

We investigate the following characteristic of a linear operator A in a Banach space X: disc[Figure not available: see fulltext.] {inf(dim R′:R′⊂R,R′εCyc A) :RεCyc A}, where Cyc A={R:R is a subspace of X, dim R<∞, span (A^{n}R: :n≥0)=X}. The value disc A is equal to the dimension of a cyclic subspace that can be chosen in an arbitrary cyclic finite dimensional subspace. If we consider a dynamical system {Mathematical expression} with the controllability property, disc A shows to what extent the dimension of the input subspace of control can be diminished without loss of controllability. In this paper we investigate when easy inequality disc A≥(the multiplicity of A) turn into the equality. Some estimates from below of disc A (of the type disc A≥sup dim Ker(A-λI)) are found for some classes of operators e.q. for compact operators, for Toeplitz operators with antianalytic symbols, for strictly lower triangular operators and some other classes.

Original language | English |
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Pages (from-to) | 274-311 |

Number of pages | 38 |

Journal | Integral Equations and Operator Theory |

Volume | 6 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Dec 1983 |

## Scopus subject areas

- Analysis
- Algebra and Number Theory