New norm inequalities for commutators of Hilbert space operators

B. Moosavi ; M. Sh. Hosseini

Problemy analiza, Tome 14 (2025) no. 1, pp. 119-129.

Résumé

New norm inequalities for commutators of Hilbert space operators are given. Among other inequalities, it is shown that if $A,$ $ B$ $\in \mathbb{B(H)}$ and there exists a real number $z_{0}$, such that $ \Vert A-z_{0}I\Vert=D_{A} $, then \begin{eqnarray*} \Vert AB \pm BA^*\Vert \leq 2 D_{A} \Vert B \Vert, \end{eqnarray*} where ${{D}_{A}}=\underset{\lambda \in \mathbb{C}}{\mathop{\inf }} \left\| A-\lambda I \right\| $. In particular, under some conditions, we prove that \begin{eqnarray*} \Vert AB\Vert \leq D_{A} \Vert B \Vert, \end{eqnarray*} which is an improvement of submultiplicative norm inequality. Also, we prove several numerical radius inequalities for products of two Hilbert space operators.

Détail

Keywords: bounded linear operator, Hilbert space, norm inequality, numerical radius.

Bibliographie

[1] Abu-Omar A., Kittaneh F., “Numerical radius inequalities for products of Hilbert space operators”, J. Operator Theory, 72:2 (2014), 521–527 | DOI | MR | Zbl

[2] Bajmaeh A. B., Omidvar M. E., “Improved Inequalities for the Extension of Euclidean Numerical Radius”, Filomat, 33:14 (2019), 4519–4524 | DOI | MR | Zbl

[3] Buzano M. L., “Generalizzazione della diseguaglianza di Cauchy-Schwarz'”, Rend. Sem. Mat. Univ. Politech.Torino, 31 (1974), 405–409 | MR | Zbl

[4] Dragomir S. S., “Rivers inequalities for the numerical radius of linear operators in Hilbert space”, Bull. Austral. Math. Soc., 73 (2006), 255–262 | DOI | MR | Zbl

[5] Dragomir S. S., “Some inequalities for the norm and the numerical radius of linear operators in Hilbert Spaces”, Tamkang J. Math., 39:1 (2008), 1–7 | DOI | MR | Zbl

[6] Dragomir S. S., “Some inequalities for commutators of bounded linear operators in a Hilbert space”, Filomat, 25:2 (2011), 151–162 | DOI | MR | Zbl

[7] Dragomir S. S., “Inequalities for the norm and the numerical radius of linear operators in Hilbert spaces”, Demonstratio Mathematica, 40:2 (2007), 411–417 | DOI | MR | Zbl

[8] Dragomir S. S., “Some inequalities for the Euclidean operator radius of two operators in Hilbert spaces”, Linear Algebra Appl., 419 (2006), 256–264 | DOI | MR | Zbl

[9] Kittaneh F., “Norm inequalities for commutators of positive operators and applications”, Mathematische Zeitschrift, 258:4 (2008), 845–849 | DOI | MR | Zbl

[10] Moosavi B., Shah Hosseini M., “Norm and numerical radius inequalities for Hilbert space operators”, The journal of analysis, 31:2 (2023), 1393–1400 | DOI | MR | Zbl

[11] Moosavi B., Shah Hosseini M., “Some inequalities for the numerical radius for operators in Hilbert $C^{*}-$modules space”, J. Inequal. Spec. Funct., 10 (2019), 77–84 | MR

[12] Omidvar M. E., Moradi H. R., “New estimates for the numerical radius of Hilbert space operators”, Linear Multilinear Algebra, 69:5 (2023), 946–956 | DOI | MR

[13] Omidvar M. E., Moradi H. R., “Better bounds on the numerical radii of Hilbert space operators”, Linear Algebra Appl., 604 (2020), 265–277 | DOI | MR | Zbl

[14] Shah Hosseini M., Moosavi B., “Inequalities for the norm and numerical radius for Hilbert $C^{*}-$module operators”, Problemy Analiza-Issues of Analysis, 9:2 (2020), 87–96 | DOI | MR | Zbl

[15] Shah Hosseini M., Moosavi B., “Some Numerical Radius Inequalities for Products of Hilbert Space Operators”, Filomat, 33:7 (2019), 2089–2093 | DOI | MR | Zbl

[16] Shah Hosseini M., Moosavi B., Moradi H. R., “An alternative estimate for the numerical radius of Hilbert space operators”, Math. Slovaca, 70:1, 233–237 | DOI | MR | Zbl

[17] Stampfli J. G., “The norm of a derivation”, Pacific J. Math., 33 (1970), 737–747 | DOI | MR | Zbl