Skip to main content
Log in

Decomposability, past and present

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

We discuss the concept of decomposability for operators from its inception to present applications.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Albrecht, E.: On decomposable operators. Integral Equations Operator Theory 2(1), 1–10 (1979)

    Article  MathSciNet  Google Scholar 

  2. Albrecht, E., Chevreau, B.: Compact perturbations of scalar type spectral operators. J. Oper. Theory 86(1), 163–188 (2021)

    Article  MathSciNet  Google Scholar 

  3. Bishop, E.: A duality theorem for an arbitrary operator. Pac. J. Math. 9, 629–642 (1959)

    Article  MathSciNet  Google Scholar 

  4. Brown, L.G.: Lidskiĭ’s theorem in the type II case. In: Geometric Methods in Operator Algebras (Kyoto, 1983), pp. 1–35. Pitman Res. Notes Math. Ser., 123. Longman Sci. Tech., Harlow (1986)

  5. Brown, S.W.: Hyponormal operators with thick spectra have invariant subspaces. Ann. of Math. (2) 125(1), 93–103 (1987)

    Article  MathSciNet  Google Scholar 

  6. Colojoară, I.: Elemente de Teorie Spectrală. Editura Academiei Republicii Socialiste România, Bucharest (1968)

    Google Scholar 

  7. Colojoară, I.: Generalized spectral operators. Rev. Math. Pures Appl. 7, 459–465 (1962)

    MathSciNet  Google Scholar 

  8. Colojoară, I., Foiaş, C.: Theory of Generalized Spectral Operators. Mathematics and its Applications, vol. 9. Gordon and Breach Science Publishers, New York-London-Paris (1968)

  9. Dunford, N.: Spectral operators. Pac. J. Math. 4, 321–354 (1954)

    Article  MathSciNet  Google Scholar 

  10. Dunford, N., Schwartz, J.T.: Linear Operators. Part III: Spectral Operators. With the assistance of William G. Bade and Robert G. Bartle. Pure and Applied Mathematics, vol. VII. Interscience Publishers [John Wiley & Sons], New York-London-Sydney (1971)

  11. Dykema, K., Haagerup, U.: Invariant subspaces of Voiculescu’s circular operator. Geom. Funct. Anal. 11(4), 693–741 (2001)

    Article  MathSciNet  Google Scholar 

  12. Dykema, K., Krishnaswamy-Usha, A.: Angles between Haagerup-Schultz projections and spectrality of operators. J. Funct. Anal. 281(4), Paper No. 109027, 26 pp. (2021)

  13. Eschmeier, J., Putinar, M.: Spectral Decompositions and Analytic Sheaves. London Mathematical Society Monographs. New Series, 10. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1996)

  14. Fang, Q., Xia, J.: Invariant subspaces for certain finite-rank perturbations of diagonal operators. J. Funct. Anal. 263(5), 1356–1377 (2012)

    Article  MathSciNet  Google Scholar 

  15. Foiaş, C.: Spectral maximal spaces and decomposable operators in Banach space. Arch. Math. (Basel) 14, 341–349 (1963)

    Article  MathSciNet  Google Scholar 

  16. Foiaş, C.: On the maximal spectral spaces of a decomposable operator. Rev. Roumaine Math. Pures Appl. 15, 1599–1606 (1970)

    MathSciNet  Google Scholar 

  17. Foiaş, C.: On the scalar parts of a decomposable operator. Rev. Roumaine Math. Pures Appl. 17, 1181–1198 (1972)

    MathSciNet  Google Scholar 

  18. Foiaş, C., Jung, I.B., Ko, E., Pearcy, C.: On rank-one perturbations of normal operators. J. Funct. Anal. 253(2), 628–646 (2007)

    Article  MathSciNet  Google Scholar 

  19. Fredholm, I., Les équations intégrales linéaires. C. R. Congr. Stockholm, 92–100 (1910)

  20. Gallardo-Gutiérrez, E., González-Doña, F.J.: Finite rank perturbations of normal operators: spectral subspaces and Borel series. J. Math. Pures Appl. 162, 23–75 (2022)

    Article  MathSciNet  Google Scholar 

  21. Haagerup, U., Schultz, H.: Invariant subspaces for operators in a general II\(_{1}\)-factor. Publ. Math. Inst. Hautes Études Sci. 109, 19–111 (2009)

    Article  MathSciNet  Google Scholar 

  22. Hilbert, D.: Grundzüge einer allgemainen Theorie der linearen Integralgleichungen. IV. Nachr. Akad. Wiss. Götingen Math. Phys. Kl., 157–227 (1906)

  23. Kapustin, V.V.: A criterion for the reflexivity of contractions with a defect operator of the Hilbert–Schmidt class. Dokl. Akad. Nauk SSSR 318(6), 1308–1311 (1991)

    MathSciNet  Google Scholar 

  24. Laursen, K.B., Neumann, M.M.: An Introduction to Local Spectral Theory. London Mathematical Society Monographs. New Series, 20. The Clarendon Press, Oxford University Press, New York (2000)

  25. Lindenstrauss, J., Tzafriri, L.: On the complemented subspaces problem. Israel J. Math. 9, 263–269 (1971)

    Article  MathSciNet  Google Scholar 

  26. Maeda, F.-Y.: Generalized spectral operators on locally convex spaces. Pac. J. Math. 13, 177–192 (1963)

    Article  MathSciNet  Google Scholar 

  27. Putinar, M.: Hyponormal operators are subscalar. J. Oper. Theory 12(2), 385–395 (1984)

    MathSciNet  Google Scholar 

  28. Putinar, M., Yakubovich, D.: Spectral dissection of finite rank perturbations of normal operators. J. Oper. Theory 85(1), 45–78 (2021)

    Article  MathSciNet  Google Scholar 

  29. Riesz, F.: Über lineare Funktionalgleichungen. Acta Math. 42, 71–98 (1918)

    Google Scholar 

  30. Taylor, J.L.: A joint spectrum for several commuting operators. J. Funct. Anal. 6, 172–191 (1970)

    Article  MathSciNet  Google Scholar 

  31. Taylor, J.L.: The analytic-functional calculus for several commuting operators. Acta Math. 125, 1–38 (1970)

    Article  MathSciNet  Google Scholar 

  32. Vasilescu, F.-H.: Analytic Functional Calculus and Spectral Decompositions. Mathematics and its Applications (East European Series), 1. D. Reidel Publishing Co., Dordrecht (1982)

    Google Scholar 

  33. Wermer, J.: On invariant subspaces of normal operators. Proc. Amer. Math. Soc. 3, 270–277 (1952)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hari Bercovici.

Additional information

Dedicated to the memory of Ciprian Foiaş on his 90th anniversary.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bercovici, H. Decomposability, past and present. Arch. Math. 121, 603–614 (2023). https://doi.org/10.1007/s00013-023-01901-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-023-01901-x

Keywords

Mathematics Subject Classification

Navigation