Advertisement

Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

On Decompositions of a Scalar Operator into a Sum of Self-Adjoint Operators with Finite Spectrum

  • 30 Accesses

We consider the problem of classification of nonequivalent representations of a scalar operator λI in the form of a sum of k self-adjoint operators with at most n 1 , . . . ,n k points in their spectra, respectively. It is shown that this problem is *-wild for some sets of spectra if (n 1 , . . . ,n k ) coincides with one of the following k -tuples: (2, . . . , 2) for k ≥ 5, (2, 2, 2, 3), (2, 11, 11), (5, 5, 5), or (4, 6, 6). It is demonstrated that, for the operators with points 0 and 1 in the spectra and k ≥ 5, the classification problems are *-wild for every rational λϵ 2 [2, 3].

This is a preview of subscription content, log in to check access.

References

  1. 1.

    P. Y. Wu, “Additive combinations of special operators,” Funct. Anal. Oper. Theory, 30, 337–361 (1994).

  2. 2.

    K. R. Davidson and L. W. Marcoux, “Linear spans of unitary and similarity orbits of a Hilbert space operator,” J. Oper. Theory, 52, 113–132 (2004).

  3. 3.

    S. Albeverio, V. Ostrovsky, and Yu. Samoilenko, “On functions on graphs and representations of a certain class of *-algebras,” J. Algebra, 308, No. 2, 567–582 (2006).

  4. 4.

    K. Matsumoto, “Self-adjoint operators as a real span of 5 projections,” Math. Jpn., 29, 291–294 (1984).

  5. 5.

    M. A. Vlasenko, A. S. Mellit, and Yu. S. Samoilenko, “On the algebras generated by linearly connected generatrices with given spectrum,” Funkts. Anal. Prilozhen., 39, No. 3, 14–27 (2005).

  6. 6.

    S. A. Kruglyak, S. V. Popovych, and Yu. S. Samoilenko, “The spectral problem and algebras associated with extended Dynkin graphs. I,” Meth. Funct. Anal. Topol., 11, No. 4, 383–396 (2005).

  7. 7.

    S. A. Kruglyak, L. A. Nazarova, and A. V. Roiter, “Orthoscalar representations of quivers in the category of Hilbert spaces,” Zap. Nauchn. Sem. POMI, 338, 180–201 (2006).

  8. 8.

    V. L. Ostrovs’kyi and Yu. S. Samoilenko, “On spectral theorems for families of linearly connected self-adjoint operators with given spectra associated with extended Dynkin graphs,” Ukr. Mat. Zh., 58, No. 11, 1556–1570 (2006); English translation: Ukr. Math. J., 58, No. 11, 1768–1785 (2006).

  9. 9.

    S. A. Kruglyak and Yu. S. Samoilenko, “On the complexity of description of representations of *-algebras generated by idempotents,” Proc. Amer. Math. Soc., 128, 1655–1664 (2000).

  10. 10.

    V. M. Bondarenko, “On connection between ⊞-decomposability and wildness for algebras generated by idempotents,” Meth. Funct. Anal. Topol., 9, No. 2, 120–122 (2003).

  11. 11.

    A. A. Kirichenko, “On linear combinations of orthoprojectors,” Uchen. Zap. Tavrich. Univ., Ser. Mat., Mekh., Informat., Kibernet., No. 2, 31–39 (2002).

  12. 12.

    K. A. Yusenko, “On quadruples of projectors connected by a linear relation,” Ukr. Mat. Zh., 58, No. 9, 1289–1295 (2006); English translation: Ukr. Math. J., 58, No. 9, 1462–1470 (2006).

  13. 13.

    V. Ostrovskyi and S. Rabanovich, “Some remarks on Hilbert representations of posets,” Meth. Funct. Anal. Topol., 19, No. 2, 149–163 (2013).

  14. 14.

    V. L. Ostrovskyi, “On *-representations of a certain class of algebras related to a graph,” Meth. Funct. Anal. Topol., 11, No. 3, 250–256 (2005).

  15. 15.

    S. A. Kruglyak, V. I. Rabanovich, and Yu. S. Samoilenko, “On sums of projections,” Funct. Anal. Appl., 36, No. 3, 182–195 (2002).

  16. 16.

    V. I. Rabanovich and Yu. S. Samoilenko, “Scalar operators representable as a sum of projectors,” Ukr. Mat. Zh., 53, No. 7, 939–952 (2001); English translation: Ukr. Math. J., 53, No. 7, 1116–1133 (2001).

  17. 17.

    S. Albeverio, K. Yusenko, D. Proskurin, and Yu. Samoilenko, “*-wildness of some classes of C*-algebras,” Meth. Funct. Anal. Topol., 12 , No. 4, 315–325 (2006).

  18. 18.

    A. S. Mellit, V. I. Rabanovich, and Yu. S. Samoilenko, “When a sum of partial reflections is equal to a scalar operator?,” Funkts. Anal. Prilozhen., 38, No. 2, 91–94 (2004).

  19. 19.

    R. V. Hrushevoi, “On conditions under which the sum of self-adjoint operators with given integer spectra is a scalar operator,” Ukr. Mat. Zh., 60, No. 4, 470–477 (2008); English translation: Ukr. Math. J., 60, No. 4, 540–550 (2008).

  20. 20.

    P. V. Omel’chenko, “On reduction of block matrices in a Hilbert space,” Ukr. Mat. Zh., 61, No. 10, 1338–1347 (2009); English translation: Ukr. Math. J., 61, No. 10, 1578–1588 (2009).

  21. 21.

    K. Yusenko and Th. Weist, “Unitarizable representations of quivers,” Algebr. Represent. Theory, 16, No. 5, 1349–1383 (2013).

Download references

Author information

Correspondence to V. I. Rabanovych.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 5, pp. 701–716, May, 2015.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Rabanovych, V.I. On Decompositions of a Scalar Operator into a Sum of Self-Adjoint Operators with Finite Spectrum. Ukr Math J 67, 795–813 (2015). https://doi.org/10.1007/s11253-015-1115-z

Download citation

Keywords

  • Hilbert Space
  • Irreducible Representation
  • English Translation
  • Scalar Operator
  • Matrix Algebra