Let \(B(H)\) denote the space of all bounded linear operators on a Hilbert space \(H\). For \(A\in B(H)\), the notations \(r(A)\) and \(w(A)\) are used for the spectral radius and the numerical radius of \(A\), respectively. Using the relation \(r(A)\leq w(A)\) and an inequality for the numerical radius of block operator matrices, the authors find a spectral radius inequality for \(r(A_1B_1+A_2B_2)\) when \(A_i,B_i\in B(H)\). This result is a generalization of some previous inequality related to the spectral radius of sums, products, and commutators of operators.
Another application of this inequality is a refinement of the well-known relation \[ \frac{1}{2}\|A\|\leq w(A)\leq\|A\|. \] The upper bound of \(w(A)\) was improved to \(\frac{1}{2}(\|A\|+\|A^2\|^{\frac{1}{2}})\) by F. Kittaneh [Stud. Math. 158, No. 1, 11–17 (2003; Zbl 1113.15302)].
We define the Aluthge transform of \(A\) as \(\tilde{A}=|A|^{\frac{1}{2}}U|A|^{\frac{1}{2}} \), where \(A=U|A|\) is the polar decomposition of \(A\). It is proved that \(w(\tilde{A})\leq w(A)\) and \(\|\tilde{A}\|\leq\|A^2\|^{\frac{1}{2}}\).
T. Yamazaki [Stud. Math. 178, No. 1, 83–89 (2007; Zbl 1114.47003)] proved that \(w(A)\leq{\frac{1}{2}} (\|A\|+w(\tilde{A}))\); here, this is improved to \(w(A)\leq {\frac{1}{2}}(\|A\|+\min_{0\leq t\leq 1}w(\tilde{A_t}))\), where \(\tilde{A_t}=|A|^t U|A|^{1-t}\) is the generalized Aluthge transform of \(A\).