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Stability of sums of operators

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Abstract

A linear operator is said to be stable (Hurwitzian) if its spectrum is located in the open left half-plane. We consider the following problem: let A and B be bounded linear operators in a Hilbert space, and A be stable. What are the conditions that provide the stability of \(A+B\)?

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References

  1. Bercovici, H., Li, W.S.: Inequalities for eigenvalues of sums in a von Neumann algebra. Recent advances in operator theory and related topics (Szeged, 1999), pp. 113–126, Oper. Theory Adv. Appl. 127, Birkhauser, Basel (2001)

  2. Daleckii, Y.L., Krein, M.G.: Stability of Solutions of Differential Equations in Banach Space. American Mathematical Society, Providence (1974)

    Google Scholar 

  3. Foth, P.: Eigenvalues of sums of pseudo-hermitian matrices. Electron. J. Linear Algebra 20, 115–125 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gil’, M.I.: Operator Functions and Localization of Spectra, Lecture Notes in Mathematics, vol. 1830. Springer, Berlin (2003)

    Book  Google Scholar 

  5. Gil’, M.I.: A bound for condition numbers of matrices. Electron. J. Linear Algebra 27, 162–171 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Helmke, U., Rosenthal, J.: Eigenvalue inequalities and Schubert calculus. Math. Nachr. 171, 207–225 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Horn, A.: Eigenvalues of sums of Hermitian matrices. Pacific J. Math. 12, 225–241 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lidskii, B.V.: Spectral polyhedron of the sum of two Hermitian matrices. Funct. Anal. Appl. 16, 139–140 (1982)

    Article  MathSciNet  Google Scholar 

  9. Miranda, H.: Diagonals and eigenvalues of sums of Hermitian matrices. Extreme cases. Proyecciones 22(2), 127–134 (2003)

    MathSciNet  Google Scholar 

  10. Peacock, M.J.M., Collings, I.B., Honig, M.L.: Eigenvalue distributions of sums and products of large random matrices via incremental matrix expansions. IEEE Trans. Inf. Theor. 54(5), 2123–2137 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Thompson, R.C., Freede, L.: On the eigenvalues of a sum of Hermitian matrices. Lin. Alg. Appl. 4, 369–376 (1971)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beer-Sheva, 84105, Israel

    Michael Gil’

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  1. Michael Gil’
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Correspondence to Michael Gil’.

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I am very grateful to the referee of this paper for his (her) really helpful remarks.

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Gil’, M. Stability of sums of operators. Ann Univ Ferrara 62, 61–70 (2016). https://doi.org/10.1007/s11565-016-0243-1

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  • DOI: https://doi.org/10.1007/s11565-016-0243-1

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