V. Kostrykin, K. A. Makarov, A. K. Motovilov
On the existence of solutions to the operator Riccati equation and the \tan\Theta theorem
(105K, LaTeX2e)

ABSTRACT.  Let A and C be self-adjoint operators such that the spectrum of A lies in a gap of the spectrum of C and let d>0 be the distance between the spectra of A and C. We prove that under these assumptions the sharp value of the constant c in the condition ||B||<cd implying the solvability of the operator Riccati equation XA-CX+XBX=B^* is equal to \sqrt{2}. We also prove an extension of the Davis-Kahan \tan\Theta theorem.