On the existence of solutions to the operator Riccati equation and the tanΘtheorem

Kostrykin, Vadim; Makarov, Konstantin A.; Motovilov, Alexander K.

On the existence of solutions to the operator Riccati equation and the tanΘtheorem

Authors

Kostrykin, Vadim

Makarov, Konstantin A.

Motovilov, Alexander K.

Description

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 \tanΘtheorem 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.
Extended version of the paper