Geodesic
The Gantmakher--Krein theorem for completely indecomposable operators in spaces of functions
Trudy Instituta matematiki, Tome 14 (2006) no. 2, pp. 73-79.

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For a completely continuous non-negative operator $A$ acting in the space $L_p(\Omega)$ or $C(\Omega)$ the existence of $k$ positive eigenvalues is proved under some additional conditions on its $j$-th $(1$ exterior power $\wedge^jA$. For the case where the operator $A$ is completely indecomposable, the simplicity of all non-zero eigenvalues is proved and the connection between the imprimitivity indices of $A$ and $\wedge^jA$ is examined. 
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O. Y. Kushel. The Gantmakher--Krein theorem for completely indecomposable operators in spaces of functions. Trudy Instituta matematiki, Tome 14 (2006) no. 2, pp. 73-79. http://geodesic.mathdoc.fr/item/TIMB_2006_14_2_a8/

[1] Gantmakher F.R., Teoriya matrits, M., 1966

[2] Gantmakher F.R., Krein M.G., Ostsillyatsionnye matritsy i yadra i malye kolebaniya mekhanicheskikh sistem, M., 1950

[3] Eveson S.P., “Eigenvalues of totally positive integral operators”, Bull. London Math. Soc., 29 (1997), 216–222 | DOI | MR | Zbl

[4] Sobolev A.V., “Abstraktnye ostsillyatsionnye operatory”, Tr. seminara po differentsialnym uravneniyam, 3, Kuibyshev, 1977, 72–78 | MR

[5] Krasnoselskii M.A., Lifshits E.A., Sobolev A.V., Pozitivnye lineinye sistemy: metod polozhitelnykh operatorov, M., 1985 | MR

[6] Karlin S., Total positivity, Stanford University Press, California, 1968 | MR

[7] Zabreiko P.P., Kushel O.Yu., Dokl. NAN Belarusi, 50:3 (2006), 9–14 | MR | Zbl

[8] Stéphanos C., “Sur une extension du calculus des substitutions linéaires”, J. Math. pures appl., 6:5 (1900), 73–128 | Zbl

[9] Ichinose T., “Spectral properties of tensor products of linear operators. I”, Transactions of the American Mathematical Society, 235 (1978), 75–113 | DOI | MR | Zbl

[10] Ichinose T., “Spectral properties of tensor products of linear operators. II”, Transactions of the American Mathematical Society, 237 (1978), 223-254 | MR | Zbl

[11] Ichinose T., “Operators on tensor products of Banach spaces”, Trans. Am. Math. Soc., 170 (1972), 197–219 | DOI | MR | Zbl

[12] Ichinose T., “Operational calculus for tensor products of linear operators in Banach spaces”, Hokkaido Math. J., 4:2 (1975), 306–334 | MR | Zbl

[13] Zabreiko P.P., Smitskikh S.V., “Ob odnoi teoreme M.G. Kreina–M.A. Rutmana”, Funktsionalnyi analiz i ego prilozheniya, 13:3 (1979), 81–82 | MR

[14] Ando T., “Positive operators in semi-ordered linear spaces”, J. Fac. Sci. Hokkaido Univ. Ser. I, 13 (1957), 214–228 | MR | Zbl

[15] Alekhno E.A., Zabreiko P.P., “Kvazipolozhitelnye elementy i nerazlozhimye operatory v idealnykh prostranstvakh. II”, Vestsi NAN Belarusi. Ser. fiz.-mat. navuk, 2003, no. 1, 5–10 | MR

[16] Stepanov G.D., “O veschestvennosti sobstvennykh znachenii integralnogo operatora, ne povyshayuschego chisla peremen znaka”, Sb. trudov aspirantov matematicheskogo fakulteta, 2, Voronezh, 1971, 43–56