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**Invariant subspaces for subscalar operators.**
*(English)*
Zbl 0651.47002

One of the famous results of S. Brown [Ann. Math. II. Ser. 125, 93- 103 (1987; Zbl 0635.47020)] says that on a Hilbert space each hyponormal operator with thick spectrum has a non-trivial invariant subspace. The proof uses a result of M. Putinar which states that each hyponormal operator is subscalar, i.e. the restriction of an operator generalized scalar in the sense of C. Foiaş. In an attempt to generalize S. Brown’s result to the context of arbitrary Banach spaces E. Albrecht and B. Chevreau proved that each restriction or quotient of a decomposable operator with sufficiently rich spectrum wich acts on a quotient of closed subspaces of \(\ell\) p \((1<p<\infty)\) has non-trivial invariant subspace [J. Oper. Theory 18, 339-372 (1987)].

In the paper under review the condition to make the Scott Brown technique work is not demanded for the space, but for the operator. The main result states that on an arbitrary Banach space each subscalar operator A with Int(\(\sigma\) (A))\(\neq \emptyset\) has a non-trivial, even rationally invariant subspace. The proof uses the Scott Brown technique together with some ideas of C. Apostol.

In the paper under review the condition to make the Scott Brown technique work is not demanded for the space, but for the operator. The main result states that on an arbitrary Banach space each subscalar operator A with Int(\(\sigma\) (A))\(\neq \emptyset\) has a non-trivial, even rationally invariant subspace. The proof uses the Scott Brown technique together with some ideas of C. Apostol.

Reviewer: J.Eschmeier

### MSC:

47A15 | Invariant subspaces of linear operators |

47B40 | Spectral operators, decomposable operators, well-bounded operators, etc. |

### Keywords:

hyponormal operator with thick spectrum; non-trivial invariant subspace; restriction or quotient of a decomposable operator; subscalar operator; rationally invariant subspace### References:

[1] | E. Albrecht andB. Chevreau, Invariant subspaces for ? p -operators having Bishop’s property (?) on a large part of their spectrum. J. Operator Theory18, 339-372 (1987). · Zbl 0655.47006 |

[2] | E.Albrecht and J.Eschmeier, Functional models and local spectral theory. Preprint. · Zbl 0881.47007 |

[3] | C. Apostol, The spectral flavour of Scott Brown’s techniques. J. Operator Theory6, 3-12 (1981). · Zbl 0479.47003 |

[4] | S. W. Brown, Some invariant subspaces for subnormal operators. Integral Equations Operator Theory1, 310-333 (1978). · Zbl 0416.47009 · doi:10.1007/BF01682842 |

[5] | S. W. Brown, Hyponormal operators with thick spectrum have invariant subspaces. Ann. of Math.125, 93-103 (1987). · Zbl 0635.47020 · doi:10.2307/1971289 |

[6] | I.Colojoara and C.Foias, Theory of generalized spectral operators. New York 1968. |

[7] | J.Eschmeier, Invariant subspaces and Bishop’s property (?). Preprint 1987. |

[8] | J.Eschmeier and M.Putinar, Bishop’s condition (?) and rich extensions of linear operators. Indiana Univ. Math. J., to appear. · Zbl 0674.47020 |

[9] | S.Lang, Real analysis. Reading, Massachusetts 1969. · Zbl 0176.00504 |

[10] | M. Putinar, Hyponormal operators are subscalar. J. Operator Theory12, 385-395 (1984). · Zbl 0573.47016 |

[11] | I.Singer, Bases in Banach spaces II. Berlin-Heidelberg-New York 1981. · Zbl 0467.46020 |

[12] | F.-H.Vasilescu, Analytic functional calculus and spectral decompositions. Dordrecht 1982. · Zbl 0495.47013 |

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