Skip to main content
SpringerLink
Log in

Multivalued Linear Operator Equation \(A^{*}A = \lambda A^{n}\)

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

The main goal of this paper is to provide a complete description of the operator solutions (eventually multivalued) of the (multivalued) operator equation \(A^{*}A = \lambda A^{n}\), where n is a positive integer, A is a closed multivalued linear operator (a closed linear relation) on a complex Hilbert space \({{\mathfrak {H}}}\) and \(\lambda \) is a complex number.

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

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Arens, R.: Operational calculus of linear relations. Pac. J. Math. 11, 9–23 (1961)

    Article  MathSciNet  Google Scholar 

  2. Coddington, E.A.: Extension Theory of Formally Normal and Symmetric Subspaces, vol. 134. Memoirs of the American Mathematical Society, Providence (1973)

    MATH  Google Scholar 

  3. Dehimi, S., Mortad, M.H., Tarcsay, Z.: On the operator equations \(A^{n} = A^{*}A\). Linear Multilinear Algebra (2019). https://doi.org/10.1080/03081087.2019.1641463

    Article  Google Scholar 

  4. Hassi, S., Sandovici, A., de Snoo, H.S.V., Winkler, H.: Form sums of nonnegative selfadjoint operators. Acta Math. Hung. 111, 81–105 (2006)

    Article  MathSciNet  Google Scholar 

  5. Laberteux, K.R.: Problem 10377. Am. Math. Mon. 101, 362 (1994)

    Google Scholar 

  6. McCullough, S.A., Rodman, L.: Hereditary classes of operators and matrices. Am. Math. Mon. 104(5), 415–430 (1997)

    Article  MathSciNet  Google Scholar 

  7. Roman, M., Sandovici, A.: The square root of nonnegative selfadjoint linear relations in Hilbert spaces. J. Oper. Theory 82(2), 357–367 (2019)

    MathSciNet  MATH  Google Scholar 

  8. Roman, M., Sandovici, A.: Adjoint to each other linear relations. Nieminen type criteria, submitted

  9. Sandovici, A.: Self-adjointness and skew-adjointness criteria involving powers of linear relations. J. Math. Anal. Appl. 470(1), 186–200 (2019)

    Article  MathSciNet  Google Scholar 

  10. Sandovici, A.: Von Neumann’s theorem for linear relations. Linear Multilinear Algebra 66(9), 1750–1756 (2018)

    Article  MathSciNet  Google Scholar 

  11. Sandovici, A.: On the adjoint of linear relations in Hilbert spaces. Mediterr. J. Math. 17, 68 (2020). https://doi.org/10.1007/s00009-020-1503-y

    Article  MathSciNet  MATH  Google Scholar 

  12. Sebestyén, Z., Tarcsay, Z.S.: On the adjoint of Hilbert space operators. Linear Multilinear Algebra (2018). https://doi.org/10.1080/03081087.2018.1430120

    Article  MATH  Google Scholar 

  13. Wang, B.Y., Zhang, F.: Words and normality of matrices. Linear Multilinear Algebra 40(2), 111–118 (1995)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Informatics, “Gheorghe Asachi”, Technical University of Iaşi, B-dul Carol I, nr. 11, 700506, Iasi, Romania

    Marcel Roman & Adrian Sandovici

Authors
  1. Marcel Roman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Adrian Sandovici
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adrian Sandovici.

Additional information

Communicated by Seppo Hassi.

Dedicated to Professor Henk de Snoo on the occasion of his 75th birthday.

Publisher's Note

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

This article is part of the topical collection “Recent Developments in Operator Theory - Contributions in Honor of H.S.V. de Snoo” edited by Jussi Behrndt and Seppo Hassi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Roman, M., Sandovici, A. Multivalued Linear Operator Equation \(A^{*}A = \lambda A^{n}\). Complex Anal. Oper. Theory 15, 6 (2021). https://doi.org/10.1007/s11785-020-01045-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11785-020-01045-8

Keywords

Mathematics Subject Classification

Search

Navigation