×

Irreducible compact operators. (English) Zbl 0607.47033

The author proves the following nice result. Let T be a positive, compact, ideal-irreducible operator in a Banach lattice E (dim \(E\geq 2)\). Then T is not quasi-nilpotent, i.e. \(r(T)>0\) where, of course, r(T) denotes the spectral radius) of T.
There is also a generalization of this result to positive irreducible operators, some powers of which majorizes a compact positive operator. This class of operators was considered by V. Caselles [Indagationes Math. 48, 11-16 (1986; Zbl 0595.47030)].
For still another generalization see the next review.
Reviewer: Yu.Abramovich

MSC:

47B60 Linear operators on ordered spaces
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
46B42 Banach lattices
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Aliprantis, C.D., Burkinshaw, O.: Positive compact operators in Banach lattices. Math. Z.174, 289-298 (1980) · Zbl 0437.46019 · doi:10.1007/BF01161416
[2] Caselles, V.: On irreducible operators on Banach lattices. Preprint (1985) · Zbl 0587.46019
[3] Dodds, P.G., Fremlin, D.H.: Compact operators in Banach lattices. Is. J. Math.34, 287-320 (1979) · Zbl 0438.47042 · doi:10.1007/BF02760610
[4] Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. Amsterdam-London: North-Holland 1971 · Zbl 0231.46014
[5] Michaels, A.J.: Hilden’s simple proof of Lomonosov’s invariant subspace theorem. Adv. Math.25, 56-58 (1977) · Zbl 0356.47003 · doi:10.1016/0001-8708(77)90089-5
[6] Schaefer, H.H.: Topologische Nilpotenz irreduzibler Operatoren. Math. Z.117, 135-140 (1970) · Zbl 0246.47001 · doi:10.1007/BF01109835
[7] Schaefer, H.H.: Banach lattices and positive operators. Berlin-Heidelberg-New York: Springer 1974 · Zbl 0296.47023
[8] Schaefer, H.H.: Some remarks on irreducible compact operators. Semesterbericht Funktional-analysis, Wintersemester 84/85, 1-8, Tübingen 1985
[9] Zaanen, A.C.: Riesz spaces II. Amsterdam-New York-Oxford: North-Holland 1983 · Zbl 0519.46001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.