Uniqueness of matrix square roots under a numerical range condition (Q5957193)

For \(A\in M_n({\mathbb C})\) denote by \(\sigma (A)\) its spectrum and by \(W(A)\) its numerical range, i.e. \(W(A)=\{ x^*Ax\mid x^*x=1\), \(x\in {\mathbb C}^n\}\). Denote by RHP the open right half of the complex plane and by \(\overline{\text{RHP}}\) its closure. The authors consider the question under what additional conditions is the square root of \(A\) unique. NEWLINENEWLINENEWLINEIt is known that if \(\sigma (A)\cap (-\infty ,0]=\emptyset\) (resp. \(W(A)\cap (-\infty ,0]=\emptyset\)), then there exists a unique \(B\in M_n({\mathbb C})\) such that \(B^2=A\) and \(\sigma (B)\subset\) RHP (resp. \(W(B)\subset\) RHP). In the paper it is shown that 1) if \(A\) is nonsingular and \(W(A)\cap (-\infty ,0)=\emptyset\), then there exists a unique \(B\in M_n({\mathbb C})\) such that \(B^2=A\) and \(W(B)\subset\) RHP; 2) if \(W(A)\cap (-\infty ,0)=\emptyset\), then there exists a unique \(B\in M_n({\mathbb C})\) such that \(B^2=A\) and \(W(B)\subset\overline{\text{RHP}}.\) NEWLINENEWLINENEWLINEFor \(2\times 2\)-matrices a complete answer to the question is given which matrix \(A\) has a unique square root \(B\) such that \(W(B)\subset\overline{\text{RHP}}.\)