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

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. Under these assumptions we prove that the best possible value of the constant c in the condition $\left\| B \right\| < cd$ 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 Θ 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 $\left\| B \right\| < d,$ and that this condition is optimal.