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Extensions of Euclidean operator radius inequalities

Mohammad Sal Moslehian, Mostafa Sattari, Khalid Shebrawi


To extend the Euclidean operator radius, we define $w_p$ for an $n$-tuple of operators $(T_1,\dots,T_n)$ in $\mathbb{B}(\mathscr{H})$ by $w_p(T_1,\dots,T_n):= \sup_{\lVert x \rVert =1} (\sum_{i=1}^{n}\lvert \langle T_i x, x \rangle \rvert^p)^{1/p}$ for $p\geq1$. We generalize some inequalities including the Euclidean operator radius of two operators to those involving $w_p$. Further we obtain some lower and upper bounds for $w_p$. Our main result states that if $f$ and $g$ are non-negative continuous functions on $[0,\infty) $ satisfying $f(t) g(t) =t$ for all $t\in [ 0,\infty) $, then \begin{equation*} w_{p}^{rp}( A_{1}^*T_{1}B_{1}, \dots ,A_{n}^*T_{n}B_{n}) \leq \frac{n^{r-1}}{2} \Bigl\lVert \sum_{i=1}^n [ B_{i}^*f^{2}( \lvert T_{i}\rvert ) B_{i}] ^{rp} + [ A_{i}^*g^{2}( \lvert T_{i}^* \rvert ) A_{i}]^{rp} \Bigr\rVert, \end{equation*} for all $p\geq 1$, $r\geq 1$ and operators in $\mathbb{B}(\mathscr{H})$.

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Dragomir, S. S., Some inequalities for the Euclidean operator radius of two operators in Hilbert spaces, Linear Algebra Appl. 419 (2006), no. 1, 256–264.

Fujii, J.-I., Fujii, M., Moslehian, M. S., Pečarić, J. E., and Seo, Y., Reverse Cauchy-Schwarz type inequalities in pre-inner product $C^ast $-modules, Hokkaido Math. J. 40 (2011), no. 3, 393–409.

Gustafson, K. E. and Rao, D. K. M., Numerical range: The field of values of linear operators and matrices, Universitext, Springer-Verlag, New York, 1997.

Hirzallah, O. and Kittaneh, F., Numerical radius inequalities for several operators, Math. Scand. 114 (2014), no. 1, 110–119.

Hirzallah, O., Kittaneh, F., and Shebrawi, K., Numerical radius inequalities for certain $2times 2$ operator matrices, Integral Equations Operator Theory 71 (2011), no. 1, 129–147.

Kittaneh, F., Notes on some inequalities for Hilbert space operators, Publ. Res. Inst. Math. Sci. 24 (1988), no. 2, 283–293.

Mitrinović, D. S., Pečarić, J. E., and Fink, A. M., Classical and new inequalities in analysis, Mathematics and its Applications (East European Series), vol. 61, Kluwer Academic Publishers Group, Dordrecht, 1993.

Pečarić, J., Furuta, T., Mićić Hot, J., and Seo, Y., Mond-Pečarić method in operator inequalities, Monographs in Inequalities: Inequalities for bounded selfadjoint operators on a Hilbert space, vol. 1, ELEMENT, Zagreb, 2005.

Popescu, G., Unitary invariants in multivariable operator theory, Mem. Amer. Math. Soc. 200 (2009), no. 941.

Sattari, M., Moslehian, M. S., and Yamazaki, T., Some generalized numerical radius inequalities for Hilbert space operators, Linear Algebra Appl. 470 (2015), 216–227.

Yamazaki, T., On upper and lower bounds for the numerical radius and an equality condition, Studia Math. 178 (2007), no. 1, 83–89.



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