On the existence of solutions to the operator Riccati equation and the tan Theta theorem

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. Under these assumptions we prove that the best possible value of the constant c in the condition parallel toBparallel to < cd guaranteeing the existence of a (bounded) solution to the operator Riccati equation XA-CX+XBX = B* is equal to root2. 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 parallel toBparallel to < d, and that this condition is optimal.