Gantmacher-Krein Theorem for 2-totally Nonnegative Operators in Ideal Spaces

  • Olga Y. Kushel
  • Petr P. Zabreiko
Part of the Operator Theory: Advances and Applications book series (OT, volume 202)


The tensor and exterior squares of a completely continuous nonnegative linear operator A acting in the ideal space X(Ω) are studied. The theorem representing the point spectrum (except, probably, zero) of the tensor square (AA) M in the terms of the spectrum of the initial operator A is proved. The existence of the second (according to the module) positive eigenvalue λ2, or a pair of complex conjugate eigenvalues of a completely continuous non-negative operator A is proved under the additional condition, that its exterior square (AA) M is also nonnegative.


Total positivity ideal spaces tensor products exterior products point spectrum 

Mathematics Subject Classification (2000)

Primary 47B65 Secondary 47A80 47B38 46E30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Boccuto, A.V. Bukhvalov, and A.R. Sambucini, Some inequalities in classical spaces with mixed norms, Positivity, 6 (2002), 393–411.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    A.V. Bukhvalov, Vector-valued function spaces and tensor products, Siberian Math. J., 13 (1972), 1229–1238 (Russian).zbMATHGoogle Scholar
  3. [3]
    -On spaces with mixed norm, Vestnik Leningrad. Univ., 19 (1973), 512 (Russian); English transl.: Vestnik Leningrad Univ. Math., 6 (1979), 303–311.Google Scholar
  4. [4]
    -Generalization of the Kolmogorov-Nagumo theorem on the tensor product, Qualitative and Approximate Methods for the Investigation of Operator Equations, 4 (1979), 4865, Jaroslavl (Russian).Google Scholar
  5. [5]
    -, Application of methods of the theory of order-bounded operators to the theory of operators in L p-spaces, Uspekhi Mat. Nauk., 38 (1983), 37–83.MathSciNetGoogle Scholar
  6. [6]
    S.P. Eveson, Eigenvalues of totally positive integral operators, Bull. London Math. Soc., 29 (1997), 216–222.CrossRefMathSciNetGoogle Scholar
  7. [7]
    F.R. Gantmacher and M.G. Krein, Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, Amer. Math. Soc., Providence, PI, 2002.Google Scholar
  8. [8]
    J.R. Holub, Compactness in topological tensor products and operator spaces, Proc. Amer. Math. Soc., 36 (1972), 398–406.CrossRefMathSciNetGoogle Scholar
  9. [9]
    T. Ichinose, Operators on tensor products of Banach spaces, Trans. Amer. Math. Soc., 170 (1972), 197–219.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    -, Operational calculus for tensor products of linear operators in Banach spaces, Hokkaido Math. J., 4 (1975), 306–334.MathSciNetGoogle Scholar
  11. [11]
    -, Spectral properties of tensor products of linear operators. I, Trans. Amer. Math. Soc., 235 (1978), 75–113.CrossRefGoogle Scholar
  12. [12]
    -, Spectral properties of tensor products of linear operators. II, Trans. Amer. Math. Soc., 237 (1978), 223–254.MathSciNetGoogle Scholar
  13. [13]
    A.S. Kalitvin, Linear operators with partial integrals, Voronezh, 2000.Google Scholar
  14. [14]
    L.V. Kantorovich and G.P. Akilov, Functional Analysis, 2nd rev. Moscow, 1977 (Russian); English transl.: Pergamon Press, Oxford, 1982.Google Scholar
  15. [15]
    S. Karlin, Total positivity, Stanford University Press, California, 1968.zbMATHGoogle Scholar
  16. [16]
    O.D. Kellog, Orthogonal function sets arising from integral equations, American Journal of Mathematics, 40 (1918), 145–154.CrossRefMathSciNetGoogle Scholar
  17. [17]
    O.Y. Kushel and P.P. Zabreiko, Gantmacher-Krein theorem for 2-nonnegative operators in spaces of functions, Abstract and Applied Analysis, Article ID 48132 (2006), 1–15.CrossRefGoogle Scholar
  18. [18]
    V.L. Levin, Tensor products and functors in categories of Banach spaces defined by KB-lineals, Trudy Moskov. Mat. Obshch., 20 (1969), 43–81 (Russian); English transl.: Trans. Moscow Math. Soc., 20 (1969), 41–78.zbMATHGoogle Scholar
  19. [19]
    T.-W. Ma, Classical analysis on normed spaces, World Scientific Publishing, 1995.Google Scholar
  20. [20]
    N.J. Nielsen, On Banach ideals determined by Banach lattices and their applications, Diss. Math. 109 (1973).Google Scholar
  21. [21]
    A. Pinkus A, Spectral properties of totally positive kernels and matrices, Total positivity and its applications, M. Gasca, C.A. Micchelli, eds., Dordrecht, Boston, London, Kluwer Acad. Publ., 1996, 1–35.Google Scholar
  22. [22]
    Yu.V. Pokornyi Yu.V., O.M. Penkin, V.L. Pryadiev, A.V. Borovskikh, K.P. Lazarev, and S.A. Shabrov, Differential equations on geometrical graphs, FIZMATLIT, Moscow, 2004.Google Scholar
  23. [23]
    A.V. Sobolev, Abstract oscillatory operators, Proceedings of the Seminar on differential equations, Kuibishev, 1977, no. 3, 72–78 (Russian).Google Scholar
  24. [24]
    V.I. Yudovich, Spectral properties of an evolution operator of a parabolic equation with one space variable and its finite-dimensional analogues, Uspekhi Mat. Nauk, 32 (1977), 230–232 (Russian).Google Scholar
  25. [25]
    P.P. Zabreiko, Ideal spaces of functions, Vestnik Yaroslav. Univ., 4 (1974), 12–52 (Russian).MathSciNetGoogle Scholar
  26. [26]
    P.P. Zabreiko and S.V. Smitskikh, A theorem of M.G. Krein and M.A. Rutman, Functional Analysis and Its Applications, 13 (1980), 222–223.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • Olga Y. Kushel
    • 1
  • Petr P. Zabreiko
    • 1
  1. 1.Department of Mechanics and MathematicsBelorussian State UniversityMinskBelarus

Personalised recommendations