Annals of Functional Analysis

Upper bounds for numerical radius inequalities involving off-diagonal operator matrices

Mojtaba Bakherad and Khalid Shebrawi

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Abstract

In this article, we establish some upper bounds for numerical radius inequalities, including those of 2×2 operator matrices and their off-diagonal parts. Among other inequalities, it is shown that if T=[0XY0], then ωr(T)2r2f2r(|X|)+g2r(|Y|)12f2r(|Y|)+g2r(|X|)12 and ωr(T)2r2f2r(|X|)+f2r(|Y|)12g2r(|Y|)+g2r(|X|)12, where X,Y are bounded linear operators on a Hilbert space H, r1, and f, g are nonnegative continuous functions on [0,) satisfying the relation f(t)g(t)=t (t[0,)). Moreover, we present some inequalities involving the generalized Euclidean operator radius of operators T1,,Tn.

Article information

Source
Ann. Funct. Anal., Volume 9, Number 3 (2018), 297-309.

Dates
Received: 6 June 2017
Accepted: 5 September 2017
First available in Project Euclid: 17 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.afa/1508205625

Digital Object Identifier
doi:10.1215/20088752-2017-0029

Mathematical Reviews number (MathSciNet)
MR3835218

Zentralblatt MATH identifier
06946355

Subjects
Primary: 47A12: Numerical range, numerical radius
Secondary: 47A30: Norms (inequalities, more than one norm, etc.) 47A63: Operator inequalities 47B33: Composition operators

Keywords
numerical radius off-diagonal part positive operator Young inequality generalized Euclidean operator radius

Citation

Bakherad, Mojtaba; Shebrawi, Khalid. Upper bounds for numerical radius inequalities involving off-diagonal operator matrices. Ann. Funct. Anal. 9 (2018), no. 3, 297--309. doi:10.1215/20088752-2017-0029. https://projecteuclid.org/euclid.afa/1508205625


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References

  • [1] A. Abu-Omar and F. Kittaneh, Estimates for the numerical radius and the spectral radius of the Frobenius companion matrix and bounds for the zeros of polynomials, Ann. Func. Anal. 5 (2014), no. 1, 56–62.
  • [2] A. Abu-Omar and F. Kittaneh, Numerical radius inequalities for $n\times n$ operator matrices, Linear Algebra Appl. 468 (2015), 18–26.
  • [3] Y. Al-manasrah and F. Kittaneh, A generalization of two refined Young inequalities, Positivity 19 (2015), no. 4, 757–768.
  • [4] J. C. Bourin, Matrix subadditivity inequalities and block-matrices, Internat. J. Math. 20 (2009), no. 6, 679–691.
  • [5] M. Hajmohamadi, R. Lashkaripour, and M. Bakherad, Some generalizations of numerical radius on off-diagonal part of $2\times2$ operator matrices, preprint, to appear in J. Math. Inequal.
  • [6] P. R. Halmos, A Hilbert Space Problem Book, 2nd ed., Grad. Texts in Math. 19, Springer, New York, 1982.
  • [7] O. Hirzallah, F. Kittaneh, and K. Shebrawi, Numerical radius inequalities for certain $2\times 2$ operator matrices, Integral Equations Operator Theory 71 (2011), no. 1, 129–149.
  • [8] F. Kittaneh, Notes on some inequalities for Hilbert space operators, Publ. Res. Inst. Math. Sci. 24 (1988), no. 2, 283–293.
  • [9] F. Kittaneh, A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math. 158 (2003), 11–17.
  • [10] K. E. Gustafson and D. K. M. Rao, Numerical Range: The Field of Values of Linear Operators and Matrices, Springer, New York, 1996.
  • [11] G. Popescu, Unitary invariants in multivariable operator theory, Mem. Amer. Math. Soc. 200 (2009), no. 941.
  • [12] G. Ramesh, On the numerical radius of a quaternionic normal operator, Adv. Oper. Theory 2 (2017), no. 1, 78–86.
  • [13] M. S. Moslehian, M. Sattari, and K. Shebrawi, Extension of Euclidean operator radius inequalities, Math Scand. 120 (2017), no. 1, 129–144.
  • [14] K. Shebrawi and H. Albadawi, Numerical radius and operator norm inequalities, J. Inequal. Appl. 2009, art. ID 492154.
  • [15] A. Sheikhhosseini, M. S. Moslehian, and Kh. Shebrawi, Inequalities for generalized Euclidean operator radius via Young’s inequality, J. Math. Anal. Appl. 445 (2017), no. 2, 1516–1529.
  • [16] T. Yamazaki, On upper and lower bounds of the numerical radius and an equality condition, Studia Math. 178 (2007), 83–89.
  • [17] A. Zamani, Some lower bounds for the numerical radius of Hilbert space operators, Adv. Oper. Theory 2 (2017), no. 2, 98–107.