Biquasitriangularity and spectral continuity. (English) Zbl 0583.47006

Lange, Ridgley

Glasg. Math. J. 26, 177-180 (1985).

Let T be an operator in the algebra L(H) of bounded operators on a Hilbert space H, and let T be a point of continuity of the map \(\sigma\) : \(A\to \sigma (A)\) (A\(\in L(H))\), where the topologies are the uniform and the Hausdorff metric, resp. The author proves that such T which is nonscalar has a proper hyperinvariant subspace or else T is the translate of a quasinilpotent operator. Another theorem gives several criteria for the biquasitriangularity of such a T. Among these are the following:
(a) \(\sigma\) (T) is nowhere dense,
(b) \(\sigma\) (T) is the closure of its isolated points, or
(c) T and \(T^*\) both have the single valued extension property.

MSC:

47A10	Spectrum, resolvent
47A66	Quasitriangular and nonquasitriangular, quasidiagonal and nonquasidiagonal linear operators

Keywords:

spectral continuity; proper hyperinvariant subspace; translate of a quasinilpotent operator; biquasitriangularity; single valued extension property

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