Numerical radius and distance from unitary operators (Q2852250)

Some notation is needed to state the main results. Namely: (i) \({\mathcal S}_\rho\) is the class of all operators \(T\) acting on a Hilbert space \(H\) for which there exist a super-space \({\mathcal H}\supset H\) and a unitary operator \(U\) acting on \({\mathcal H}\) such that NEWLINE\[NEWLINE T^n=\rho PU^nP^*, \quad n=1,2,\dots,NEWLINE\]NEWLINE where \(P\) is the orthogonal projection of \({\mathcal H}\) onto \(H\); (ii) the operator \(\rho\)-radius of a bounded linear operator \(A\) acting on \(H\) is NEWLINE\[NEWLINE w_\rho(A)=\inf\{\lambda>0: \lambda^{-1}A\in{\mathcal S}_\rho\}, NEWLINE\]NEWLINE and (iii) \(\psi_\rho(r)\) for \(r\geq 1\) is defined as \(\sup\{ \|A\|: w_\rho(A)\leq r, w_\rho(A^{-1})\leq r\}\).NEWLINENEWLINEFor \(\rho\in [1,2]\), it is shown in the paper that \(\psi_\rho(r)\leq X_\rho(r)+\sqrt{X_\rho(r)^2-1}\), where NEWLINE\[NEWLINE X_\rho(r)=\frac{2+\rho r^2-\rho+\sqrt{(2+\rho r^2-\rho)^2-4r^2}}{2r}, NEWLINE\]NEWLINE and that NEWLINE\[NEWLINE \psi_\rho(1+\epsilon)\leq 1+\root 4\of{8(\rho-1)\varepsilon}+O(\varepsilon^{1/2}), \quad \varepsilon\to 0.\tag{1}NEWLINE\]NEWLINE The case \(\rho=2\) in which \(w_2(A)\) coincides with the numerical radius \(w(A)\) of \(A\) while \(X_2(r)\) simplifies to \(X(r)=r+\sqrt{r^2-1}\), is stated separately. It is also observed that in this case the exponent \(1/4\) in (1) is optimal.