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Integral Representations and Sharp Numerical Radius Inequalities for Hilbert Space Operators
We study the numerical radius \(\omega(T)\) of a bounded linear operator \(T\) on a complex Hilbert space. Write \(T=A+iB\), where \(A\) and \(B\) are the real and imaginary parts of \(T\). A central result is the exact formula \[ \omega(T)=\frac{\pi}{2}\sup_{\|x\|=1}\int_0^1 \left|\left\langle (A\cos \pi t+B\sin \pi t...