## Nihonkai Mathematical Journal

### Dunkl-Williams inequality for operators associated with $p$-angular distance

#### Abstract

We present several operator versions of the Dunkl--Williams inequality with respect to the $p$-angular distance for operators. More precisely, we show that if $A, B \in \mathbb{B}(\mathscr{H})$ such that $|A|$ and $|B|$ are invertible, $\frac{1}{r}+\frac{1}{s}=1$ $(r>1)$ and $p\in\mathbb{R}$, then

\begin{equation*} \left|\,A|A|^{p-1}-B|B|^{p-1}\,\right|^{2} \leq |A|^{p-1}\left(\,r|A-B|^{2}+s\left|\,|A|^{1-p}|B|^{p}-|B|\,\right|^2\,\right)|A|^{p-1}.%\nonumber \end{equation*}

In the case that $0 < p \leq 1$, we remove the invertibility assumption and show that if $A=U|A|$ and $B=V|B|$ are the polar decompositions of $A$ and $B$, respectively, $t>0$, then

$$\left|\,\left(U|A|^{p}-V|B|^{p}\right)|A|^{1-p}\,\right|^{2}\leq \left(1+t\strut\right)|A-B|^{2}+\left(1+\frac{1}{t}\right) \left| |B|^{p}|A|^{1-p}-|B| \right|^2 .$$

We obtain several equivalent conditions, when the case of equalities hold.

#### Article information

Source
Nihonkai Math. J., Volume 21, Number 1 (2010), 11-20.

Dates
First available in Project Euclid: 8 April 2011

https://projecteuclid.org/euclid.nihmj/1302268213

Mathematical Reviews number (MathSciNet)
MR2798091

Zentralblatt MATH identifier
1225.47020

Subjects
Primary: 47A63: Operator inequalities
Secondary: 26D15: Inequalities for sums, series and integrals

#### Citation

Dadipour, Farzad; Fujii, Masatoshi; Moslehian, {Mohammad Sal. Dunkl-Williams inequality for operators associated with $p$-angular distance. Nihonkai Math. J. 21 (2010), no. 1, 11--20. https://projecteuclid.org/euclid.nihmj/1302268213

#### References

• S. S. Dragomir, Inequalities for the $p$-angular distance in normed linear spaces, Math. Inequal. Appl., 12 (2009), 391–401.
• S. S. Dragomir, Generalization of the Pečarić-Rajić inequality in normed linear spaces, Math. Inequal. Appl., 12 (2009), 53–65.
• C. F. Dunkl and K. S. Williams, A Simple Norm Inequality, Amer. Math. Monthly, 71 (1964), 53–54.
• T. Furuta, Invitation to linear operators. From matrices to bounded linear operators on a Hilbert space, Taylor & Francis, Ltd., London, 2001.
• O. Hirzallah, Non-commutative operator Bohr inequality, J. Math. Anal. Appl., 282 (2003), 578–583.
• A. Jiménez-Melado, E. Llorens-Fuster and E. M. Mazcuñán-Navarro, The Dunkl–Williams constant, convexity, smoothness and normal structure, J. Math. Anal. Appl., 342 (2008), 298–310.
• W. A. Kirk and M. F. Smiley, Mathematical Notes: Another characterization of inner product spaces, Amer. Math. Monthly, 71 (1964), 890–891.
• L. Maligranda, Simple norm inequalities, Amer. Math. Monthly, 113 (2006), 256–260.
• P. P. Mercer, The Dunkl–Williams inequality in an inner product space, Math. Inequal. Appl., 10 (2007), 447–450.
• J. G. Murphy, $C^*$-Algebras and Operator Theory, Academic Press, San Diego, 1990.
• J. E. Pečarić and R. Rajić, The Dunkl–Williams inequality with $n$-elements in normed linear spaces, Math. Inequal. Appl., 10 (2007), 461–470.
• J. E. Pečarić and R. Rajić, Inequalities of the Dunkl–Williams type for absolute value operators, J. Math. Inequal., 4 (2010), 1–10.
• K.-S. Saito and M. Tominaga, The Dunkl-Williams type inequality for absolute value operators, Linear Algebra Appl., 432 (2010), 3258–3264.