Some inequalities for the Euclidean operator radius of two operators in Hilbert spaces
The Euclidean operator radius of a pair of bounded linear operators \(A\) and \(B\) on a Hilbert space \(H\) with inner product \(\langle\cdot,\cdot\rangle\) is \(w_e(A, B)= \sup_{\| x\|= 1}(|\langle Ax, x\rangle|^2+ |\langle Bx, x\rangle|^2)^{1/2}\), as was first defined by \textit{G.\,Popescu} [``Unitary invariants in multivariable operator theory'' (Preprint) (2004; arXiv:math/0410492)]. For selfadjoint \(A\) and \(B\), \(W_e(A, B)\) reduces to the numerical radius \(w(A+ iB)\) of \(A+ iB\). In the present paper, the author proves miscellaneous sharp inequalities involving the Euclidean operator radii, numerical radii and norms of various operators, which generalize some previous ones by \textit{F.\,Kittaneh} [Stud.\ Math.\ 168, No.\,1, 73--80 (2005; Zbl 1072.47004)] and Popescu. For example, it is shown that \(w_e(A, B)\geq (\sqrt{2}/2) w(A^2+ B^2)^{1/2}\) and, as a consequence, \(w(C)^2\geq w(\alpha^2 C^2+ \beta^2 C^{*2})/(2|\alpha|^2+ 2|\beta|^2)\) for any operator \(C\) and scalars \(\alpha\) and \(\beta\) not both zero. On the other hand, some sharp upper bounds for \(w_e(A, B)^2\) are shown to be \(\max\{\| A\|^2,\| B\|^2\}+ w(B^*A)\) and \([\| A^*A+ B^*B\|+\| A^*A- B^*B\|]/2+ w(B^* A)\) and, from the former, one for \(w(C)^2\) is \([\max\{\| C+ C^*\|^2,\| C- C^*\|^2\}+ w((C^*- C)(C+ C^*))]/4\). The proofs for these are all elementary.
scientific article; zbMATH DE number 5077737
scientific article