Annals of Functional Analysis

A Grüss type operator inequality

T. Bottazzi and C. Conde

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In 2001, Renaud obtained a Grüss type operator inequality involving the usual trace functional. In this article, we give a refinement of that result, and we answer positively Renaud’s open problem.

Article information

Ann. Funct. Anal., Volume 8, Number 1 (2017), 124-132.

Received: 5 May 2016
Accepted: 25 July 2016
First available in Project Euclid: 12 November 2016

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Zentralblatt MATH identifier

Primary: 39B05: General
Secondary: 39B42: Matrix and operator equations [See also 47Jxx] 47B10: Operators belonging to operator ideals (nuclear, p-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20] 47A12: Numerical range, numerical radius 47A30: Norms (inequalities, more than one norm, etc.)

Grüss inequality variance trace inequality distance formula


Bottazzi, T.; Conde, C. A Grüss type operator inequality. Ann. Funct. Anal. 8 (2017), no. 1, 124--132. doi:10.1215/20088752-3764566.

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  • [1] K. Audenaert, Variance bounds, with an application to norm bounds for commutators, Linear Algebra Appl. 432 (2010), no. 5, 1126–1143.
  • [2] G. Björck and V. Thomée, A property of bounded normal operators in Hilbert space, Ark. Mat. 4 (1963), 551–555.
  • [3] S. Dragomir, Some refinements of Schwarz inequality, Suppozionul de Matematică şi Aplica ţii, Polytechnical Institute Timişoara, Romania, 1–2, (1985), 13–16.
  • [4] L. Gevorgyan, On minimal norm of a linear operator pencil, Dokl. Nats. Akad. Nauk Armen. 110 (2010), no. 2, 97–104.
  • [5] I. Gohberg and M. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr. 18, Amer. Math. Soc., Providence, R.I., 1969.
  • [6] G. Grüss, Über das Maximum des absoluten Betrages von $\frac{1}{b-a}\int_{a}^{b}f(x)g(x)\,dx-\frac{1}{(b-a)^{2}}\int_{a}^{b}f(x)\,dx\int_{a}^{b}g(x)\,dx$, Math. Z., 39 (1935), 215–226.
  • [7] V. Istratescu, On a class of normaloid operators, Math. Z. 124 (1972), 199–202.
  • [8] J. S. Matharu and M. S. Moslehian, Grüss inequality for some types of positive linear maps, J. Operator Theory 73 (2015), no. 1, 265-278.
  • [9] K. Paul, Translatable radii of an operator in the direction of another operator, Sci. Math. 2 (1999), no. 1, 119–122.
  • [10] S. Prasanna, The norm of a derivation and the Björck-Thomeé-Istr$\breve{a}$ţescu theorem, Math. Japon. 26 (1981), no. 5, 585–588.
  • [11] P. Renaud, A matrix formulation of Grüss inequality, Linear Algebra Appl. 335 (2001), 95–100.
  • [12] J. Stampfli, The norm of a derivation, Pacific J. Math. 33 (1970), 737–747.