Unitarily invariant norm inequalities for functions of accretive-dissipative \(2\times 2\) block matrices


Let \(T_{11},T_{12},T_{21},\) and \(T_{22}\) be \(n\times n\) complex matrices, \(\ T=\left( \begin{array}{cc} T_{11} &{} T_{12} \\ T_{21} &{} T_{22} \end{array} \right) \) be accretive-dissipative, \(\gamma \in (0,1]\), \(r\ge 2,\ \)and let \( \alpha ,\beta \in [0,1]\) such that \(\alpha +\beta =1\). If f is an increasing convex function on \([0,\infty )\) such that \(f(0)=0\), then

$$\begin{aligned} \left| \left| \left| f\left( \left| T_{12}+\left( 2\alpha -1\right) T_{21}^{*}\right| ^{r}\right) +f\left( 2^{r}\alpha ^{r/2}\beta ^{r/2}\left| T_{21}^{*}\right| ^{r}\right) \right| \right| \right| \le \gamma \left| \left| \left| f\left( \frac{\left| T\right| ^{r}}{\gamma ^{r/2}}\right) \right| \right| \right| \end{aligned}$$

for every unitarily invariant norm. In addition, if T is contraction and f is submultiplicative, then

$$\begin{aligned}&\left| \left| \left| \left( f\left( \left| T_{12}+\left( 2\alpha -1\right) T_{21}^{*}\right| ^{2}\right) +f\left( 4\alpha \beta \left| T_{21}^{*}\right| ^{2}\right) \right) \right| \right| \right| \\&\quad \le f^{2}\left( \sqrt{2}\right) \left| \left| \left| f^{r}\left( \left| T_{11}\right| ^{1/2}\right) \right| \right| \right| ^{1/r}\left| \left| \left| f^{s}\left( \left| T_{22}\right| ^{1/2}\right) \right| \right| \right| ^{1/s} \end{aligned}$$

for every unitarily invariant norm and for every positive real numbers rs with \(\frac{1}{r}+\frac{1}{s}=1\). Related inequalities for concave functions are also given.

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  1. 1.

    Aujla, J.S., Silva, F.C.: Weak majorization inequalities and convex functions. Linear Algebra Appl. 369, 217–233 (2003)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bhatia, R.: Matrix Analysis. Springer, New York (1997)

    Google Scholar 

  3. 3.

    Bhatia, R., Holbrook, J.: On the Clarkson-McCarthy inequalities. Math. Ann. 281, 7–12 (1988)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bourin, J.-C., Uchiyama, M.: A matrix subadditivity inequality for \(f(A+B)\) and \(f(A)+f(B)\). Linear Algebra Appl. 423, 512–518 (2007)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Fan, K., Hoffman, A.J.: Some metric inequalities in the space of matrices. Proc. Am. Math. Soc. 6, 111–116 (1955)

    MathSciNet  Article  Google Scholar 

  6. 6.

    George, A., Ikramov, KhD: On the properties of accretive-dissipative matrices. Math. Notes 77, 767–776 (2005)

    MathSciNet  Article  Google Scholar 

  7. 7.

    George, A., Ikramov, KhD, Kucherov, A.B.: On the growth factor in Gaussian elimination for generalized Higham matrices. Numer. Linear Algebra Appl. 9, 107–114 (2002)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Gohberg, I.C., Krein, M.G.; Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monographs, Vol. 18, Amer. Math. Soc., Providence, RI (1969)

  9. 9.

    Gunzburger, M.D., Plemmons, R.J.: Energy conserving norms for the solution of hyperbolic systems of partial differential equations. Math. Comput. 33, 1–10 (1979)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Gumus, I.H., Hirzallah, O., Kittaneh, F.: Norm inequalities involving accretive-dissipative \(2\times 2\) matrices. Linear Algebra Appl. 528, 76–93 (2017)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Higham, N.J.: Factorizing complex symmetric matrices with positive real and imaginary parts. Math. Comput. 67, 1591–1599 (1998)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Hirzallah, O., Kittaneh, F.: Non-commutative Clarkson inequalities for \(n\)-tuples of operators. Integral Equ. Oper. Theory 60, 369–379 (2008)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  14. 14.

    Kosem, T.: Inequalities between \(||(A+B)||\) and \(||(A)+f(B)||\). Linear Algebra Appl. 418, 153–160 (2006)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Lin, M.: Reverse determinantal inequalities for accretive-dissipative matrices. Math. Inequal. Appl. 12, 955–958 (2012)

    MATH  Google Scholar 

  16. 16.

    Lin, M.: Fischer type determinantal inequalities for accretive dissipative matrices. Linear Algebra Appl. 438, 2808–2812 (2013)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Lin, M.: A note on the growth factor in Gaussian elimination for accretive-dissipative matrices. Calcolo 51, 363–366 (2014)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Lin, M., Zhou, D.: Norm inequalities for accretive-dissipative matrix matrices. J. Math. Anal. Appl. 407, 436–442 (2013)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Siegel, C.L.: Topics in Complex Function Theory, vol. III. Wiley, New York (1973)

    Google Scholar 

  20. 20.

    Tao, Y.: More results on singular value inequalities of matrices. Linear Algebra Appl. 416, 724–729 (2006)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Zhan, X.: Matrix Theory, Grad. Stud. Math., Vol. 147, Amer. Math. Soc., Providence, RI (2013)

  22. 22.

    Zhang, Y.: Unitarily invariant norm inequalities for accretive-dissipative operators. J. Math. Anal. Appl. 412, 564–569 (2014)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Uchiyama, M.: Subadditivity of eigenvalue sums. Proc. Am. Math. Soc. 134, 1405–1412 (2006)

    MathSciNet  Article  Google Scholar 

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The authors are grateful to the referee for his comments and suggestions.

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Correspondence to Fuad Kittaneh.

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Bourahli, A., Hirzallah, O. & Kittaneh, F. Unitarily invariant norm inequalities for functions of accretive-dissipative \(2\times 2\) block matrices. Positivity (2020). https://doi.org/10.1007/s11117-020-00770-w

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  • Accretive-dissipative matrix
  • Concave function
  • Convex function
  • Contraction
  • Schatten p-norm
  • Singular value
  • Unitarily invariant norm

Mathematics Subject Classification

  • 15A18
  • 15A60
  • 47A30
  • 47B15