Skip to main content
SpringerLink
Log in

On the Decomposition of an Operator into a Sum of Four Idempotents

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We prove that operators of the form (2 ± 2/n)I + K are decomposable into a sum of four idempotents for integer n > 1 if there exists the decomposition K = K 1 ⊕ K 2 ⊕ ... ⊕ K n, \(\sum\nolimits_1^n {K_i = 0} \), of a compact operator K. We show that the decomposition of the compact operator 4I + K or the operator K into a sum of four idempotents can exist if K is finite-dimensional. If n tr K is a sufficiently large (or sufficiently small) integer and K is finite-dimensional, then the operator (2 − 2/n)I + K [or (2 + 2/n)I + K] is a sum of four idempotents.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. C. Pearcy and D. M. Topping, “Sums of small number of idempotents,” TIMiich. Math. J.,14, 453–465 (1967).

    Google Scholar 

  2. J.-H. Wang, Decomposition of Operators into Quadratic Type,Ph. D. Dissertation, National Chiao Tung University, Hsinchu, Taiwan (1991).

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

    Google Scholar 

  4. Yu. V. Bespalov, “Collections of orthoprojectors connected by relations,” Ukr. Mat. Zh.,44, No. 3, 309–317 (1992).

    Google Scholar 

  5. S. A. Kruglyak, V. I. Rabanovich, and Yu. S. Samoilenko, “Sums of projectors,” Funkts. Anal. Prilozhen.,36, No. 3, 20–35(2002).

    Google Scholar 

  6. A. Brown, P. S. Halmos, and C. Pearcy, “Commutators of operators on Hilbert space,” Can. J. Math.,17, 695–708 (1965).

    Google Scholar 

  7. C. Laurier, B. Mathes, and H. Radjavi, “Sums of idempotents,” Linear Algebra Appl.,208/209, 175–197 (1994).

    Google Scholar 

  8. P. A. Filmore, J. G. Stampfli, and J. P. Williams, “On the essential numerical range, the essential spectrum, and a problem of Halmos,” Acta Sci. Math. (Szeged), 33, 179–192 (1972).

    Google Scholar 

  9. J.-H. Wang and P. Y. Wu, “Difference and similarity models of two idempotent operators,” Linear Algebra Appl.,208/209,257–282 (1994).

    Google Scholar 

  10. N. Krupnik, S. Rosh, and B. Silbermann, “On TIC??-algebras generated by idempotents,” J. Funct. Anal.,137, 303–319 (1996).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv

    V. I. Rabanovych

Authors
  1. V. I. Rabanovych
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rabanovych, V.I. On the Decomposition of an Operator into a Sum of Four Idempotents. Ukrainian Mathematical Journal 56, 512–519 (2004). https://doi.org/10.1023/B:UKMA.0000045693.23941.9a

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:UKMA.0000045693.23941.9a

Keywords

Search

Navigation