Cartesian decomposition and Numerical radius inequalities

By F. Kıttaneh, Mohammad Sal Moslehian, T. Yamazaki. 2015

99 citations recorded by the source corpus.

We show that if $T=H+iK$ is the Cartesian decomposition of $T\in \mathbb{B(\mathscr{H})}$, then for $\alpha ,\beta \in \mathbb{R}$, $\sup_{\alpha ^{2}+\beta ^{2}=1}\Vert \alpha H+\beta K\Vert =w(T)$. We then apply it to prove that if $A,B,X\in \mathbb{B(\mathscr{H})}$ and $0\leq mI\leq X$, then \begin{align*} m\Vert \mbox{Re}(A)-\mbox{Re}(B)\Vert & \leq w(\mbox{Re}(A)X-X\mbox{Re}(B)) \\ & \leq \frac{1}{2}\sup_{\theta \in \mathbb{R}}\left\Vert (AX-XB)+e^{i\theta }(XA-BX)\right\Vert \\ & \leq \frac{\Vert AX-XB\Vert +\Vert XA-BX\Vert }{2}, \end{align*} where $\mbox{Re}(T)$ denotes the real part of an operator $T$. A refinement of the triangle inequality is also shown.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ecba91965071c10fbc515f593e6bf3a1b1c028ee

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In Applied Mathematics

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Numerical radius inequalities and its applications in estimation of zeros of polynomials (2019, unknown): “[11] proved that w ( T ) ≥ k Re ( T ) k and w ( T ) ≥ k Im ( T ) k .”
A note on the A-numerical radius of operators in semi-Hilbert spaces (2020, background): “If A = I, we get the classical numerical radius (see [5,8,9]).”
Seminorm and numerical radius inequalities of operators in semi-Hilbertian spaces (2019, background): “Some other related topics can be found in [1, 2, 3, 4, 8, 9, 10, 11, 12, 13, 15, 17].”
A-Numerical Radius Orthogonality and Parallelism of Semi-Hilbertian Space Operators and Their Applications (2020, methodology): “in [16] and it is used in the proof of our next result.”
On inequalities for A-numerical radius of operators (2019, background): “ON INEQUALITIES FOR A-NUMERICAL RADIUS OF OPERATORS 3 Over the years many mathematicians have studied numerical radius inequalities in [5, 7, 8, 9, 13, 14, 15, 16, 17, 18, 21, 22, 23, 24].”
A-numerical radius inequalities for semi-Hilbertian space operators (2019, unknown): “In Section 2, inspired by the numerical radius inequalities of bounded linear operators in [1], [2], [21], [23], [33] and by using some ideas of them, we first state a useful characterization of the A-numerical radius for T ∈ BA(H) as follows: wA(T ) = sup α(2)+β(2)=1 ∥ ∥ ∥ ∥ α T + T ♯A 2 + β T − T ”