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

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dc.creatorKostrykin, Vadim
dc.creatorMakarov, Konstantin A.
dc.creatorMotovilov, Alexander K.
dc.date2002-10-02
dc.date2003-05-29
dc.date.accessioned2026-07-07T04:51:27Z
dc.date.available2026-07-07T04:51:27Z
dc.descriptionLet 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.
dc.descriptionExtended version of the paper
dc.identifierhttps://arxiv.org/abs/math/0210032
dc.identifierhttp://arxiv.org/abs/math/0210032
dc.identifierIntegral Equations and Operator Theory 51 (2005), 121--140
dc.identifierdoi:10.1007/s00020-003-1248-6
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/65150
dc.subjectSpectral Theory
dc.subjectMathematical Physics
dc.subjectFunctional Analysis
dc.subject47A15; 39B42; 47A56
dc.titleOn the existence of solutions to the operator Riccati equation and the tanΘtheorem
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