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Let A and C be self-adjoint operators such that the spectrum of A lies in agap of the spectrum of C and let d>0 be the distance between the spectra of Aand C. We prove that under these assumptions the sharp value of the constant cin the condition ||B||<cd guaranteeing the existence of a (bounded) solution tothe operator Riccati equation XA-CX+XBX=B^* is equal to \sqrt{2}. We also provean extension of the Davis-Kahan \tan\Theta theorem and provide a sharp estimatefor the norm of the solution to the Riccati equation. If C is bounded, weprove, in addition, that the solution X is a strict contraction if B satisfiesthe condition ||B||<d, and that this condition is sharp.Comment: Extended version of the pape

On the Existence of Solutions to the Operator Riccati Equation and the tan ? Theorem