TY - JOUR

T1 - On rank one perturbations of selfadjoint operators

AU - Hassi, Seppo

AU - deSnoo, H

N1 - Relation: http://www.rug.nl/informatica/organisatie/overorganisatie/iwi
Rights: University of Groningen. Research Institute for Mathematics and Computing Science (IWI)

PY - 1997/11

Y1 - 1997/11

N2 - Let A be a selfadjoint operator in a Hilbert space h. Its rank one perturbations A+τ(·,ω)ω, τ ∈ R, are studied when ω belongs to the scale space h-2 associated with h+2 = dom A and (·,·) is the corresponding duality. If A is nonnegative and ω belongs to the scale space h-1, it is proven that the spectral measures of A(τ), τ ∈ R, converge weakly to the spectral measure of the limiting perturbation A(∞). In fact A(∞) can be identified as a Friedrichs extension. Further results for nonnegative operators A were obtained by allowing ω ∈ h-2. Our purpose is to show that most of these results are valid for rank one perturbations of selfadjoint operators, which are not necessarily semibounded. We use the fact that rank one perturbations constitute selfadjoint extensions of an associated symmetric operator. The use of so-called Q-functions facilitates the descriptions. In the special case that ω belongs to the scale space h-1 associated with h+1 = dom |A|^½, the limiting perturbation A(∞) is shown to be the generalized Friedrichs extension.

AB - Let A be a selfadjoint operator in a Hilbert space h. Its rank one perturbations A+τ(·,ω)ω, τ ∈ R, are studied when ω belongs to the scale space h-2 associated with h+2 = dom A and (·,·) is the corresponding duality. If A is nonnegative and ω belongs to the scale space h-1, it is proven that the spectral measures of A(τ), τ ∈ R, converge weakly to the spectral measure of the limiting perturbation A(∞). In fact A(∞) can be identified as a Friedrichs extension. Further results for nonnegative operators A were obtained by allowing ω ∈ h-2. Our purpose is to show that most of these results are valid for rank one perturbations of selfadjoint operators, which are not necessarily semibounded. We use the fact that rank one perturbations constitute selfadjoint extensions of an associated symmetric operator. The use of so-called Q-functions facilitates the descriptions. In the special case that ω belongs to the scale space h-1 associated with h+1 = dom |A|^½, the limiting perturbation A(∞) is shown to be the generalized Friedrichs extension.

KW - SELF-ADJOINT EXTENSIONS

M3 - Article

SN - 0378-620X

VL - 29

SP - 288

EP - 300

JO - Integral equations and operator theory

JF - Integral equations and operator theory

IS - 3

ER -