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 guaranteeing the existence of a (bounded) solution to
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 and provide a sharp estimate
for the norm of the solution to the Riccati equation. If C is bounded, we
prove, in addition, that the solution X is a strict contraction if B satisfies
the condition ||B||<d, and that this condition is sharp.Comment: Extended version of the pape