A generalization of the $tan 2Θ$ Theorem
| dc.creator | Kostrykin, Vadim | |
| dc.creator | Makarov, Konstantin A. | |
| dc.creator | Motovilov, Alexander K. | |
| dc.date | 2003-02-03 | |
| dc.date.accessioned | 2026-07-07T04:54:52Z | |
| dc.date.available | 2026-07-07T04:54:52Z | |
| dc.description | Let $\mathbf{A}$ be a bounded self-adjoint operator on a separable Hilbert space $\mathfrak{H}$ and $\mathfrak{H}_0\subset\mathfrak{H}$ a closed invariant subspace of $\mathbf{A}$. Assuming that $\sup\spec(A_0)\leq \inf\spec(A_1)$, where $A_0$ and $A_1$ are restrictions of $\mathbf{A}$ onto the subspaces $\mathfrak{H}_0$ and $\mathfrak{H}_1=\mathfrak{H}_0^\perp$, respectively, we study the variation of the invariant subspace $\mathfrak{H}_0$ under bounded self-adjoint perturbations $\mathbf{V}$ that are off-diagonal with respect to the decomposition $\mathfrak{H} = \mathfrak{H}_0\oplus\mathfrak{H}_1$. We obtain sharp two-sided estimates on the norm of the difference of the orthogonal projections onto invariant subspaces of the operators $\mathbf{A}$ and $\mathbf{B}=\mathbf{A}+\mathbf{V}$. These results extend the celebrated Davis-Kahan $\tan 2Θ$ Theorem. On this basis we also prove new existence and uniqueness theorems for contractive solutions to the operator Riccati equation, thus, extending recent results of Adamyan, Langer, and Tretter. | |
| dc.identifier | https://arxiv.org/abs/math/0302020 | |
| dc.identifier | http://arxiv.org/abs/math/0302020 | |
| dc.identifier | Operator Theory: Advances and Applications, Vol. 149, 2004, pp. 349 - 372. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/66427 | |
| dc.subject | Spectral Theory | |
| dc.subject | 47A15, 47A55, 47A62 (Primary); 47A53 (Secondary) | |
| dc.title | A generalization of the $tan 2Θ$ Theorem | |
| dc.type | text |