Unitary solutions of the equation \(cu+u^*c=2d\) (Q1124944)

Let \({\mathcal A}\) be a unital \(C^*\)-algebra with unit \(e\). We recall that every \(C^*\)-algebra may be regarded as a closed subalgebra of the set \(B(H)\) of the bounded linear operators on a Hilbert space \(H\). Supposing that \(c,d\in {\mathcal A}\) are self-adjoint with \(c\) invertible, conditions are derived for the existence of unitary solutions \(u\) of the operator equation (1) \(cu+u^*c=2d\). In particular, the fulfillment of the inequality \(\lambda^2 e+2\lambda d+c^2\geq ke\) for each real \(\lambda\), where \(k\) is a positive constant, guarantees existence of unitary solutions to (1). The closely related Riccati equation \(-X^2+B^* X+XB+A=0\) in \(B(H)\) is also studied. Reviewer's remark: Some of the results presented may be expressed in the framework of the theory of operator circles, see e.g. the excellent book of \textit{I. M. Glazman} and \textit{Yu. I. Lyubich} [Finite-dimensional linear analysis. ``Nauka'', Moscow (1969; Zbl 0332.15001) p. 441-442] or the paper of \textit{M. M. Konstantinov}, \textit{M. O. Stanislavova} and \textit{P. Hr. Petkov} [Linear Algebra Appl. 285, No. 1-3, 7-31 (1998; Zbl 0933.15025)].