Perturbation of spectra and spectral subspaces

We consider the problem of variation of spectral subspaces for linear self-adjoint operators under off-diagonal perturbations. We prove a number of new optimal results on the shift of the spectrum and obtain (sharp) estimates on the norm of the difference of two spectral projections

On a Subspace Perturbation Problem

We discuss the problem of perturbation of spectral subspaces for linear self-adjoint operators on a separable Hilbert space. Let $A$ and $V$ be bounded self-adjoint operators. Assume that the spectrum of $A$ consists of two disjoint parts $\sigma$ and $\Sigma$ such that $d=\text{dist}(\sigma, \Sigma)>0$. We show that the norm of the difference of the spectral projections \EE_A(\sigma) and \EE_{A+V}\big (\{\lambda | \dist(\lambda, \sigma) $<d/2\}\big)$ for $A$ and $A+V$ is less then one whenever either (i) $\|V\|<\frac{2}{2+\pi}d$ or (ii) $\|V\|<{1/2}d$ and certain assumptions on the mutual disposition of the sets $\sigma$ and $\Sigma$ are satisfied

Property control methods of diamond-like silicon-carbon films for micro- and nanoelectronics

Possible methods for controlling the properties of amorphous diamond-like silicon-carbon films are considered: physical or structural modification, chemical modification, and physical-chemical modification. It is shown that the method of physical modification allows controlling in a wide range the properties of diamondlike silicon-carbon films (electrophysical, mechanical properties and surface morphology) without changing the chemical composition of the material. Chemical modification was carried out by introducing transition metal into diamond-like silicon-carbon films. The dependences of the phase composition, electrophysical and mechanical properties on the content and type of metal are analyzed. The method of physical-chemical modification is considered, when the introduced impurity changes not only the chemical composition, but also the structure of the material

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. 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 \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 ||B||<d, and that this condition is sharp.Comment: Extended version of the pape