A Krein-Like Formula for Singular Perturbations of Self-Adjoint Operators and Applications
Abstract
Description
Given a self-adjoint operator $A:D(A)\subseteq\calH\to\calH$ and a continuous linear operator $τ:D(A)\to\X$ with Range$ τ'\cap\calH' ={0}$, $\X$ a Banach space, we explicitly construct a family $A^τ_Θ$ of self-adjoint operators such that any $A^τ_Θ$ coincides with the original $A$ on the kernel of $τ$. Such a family is obtained by giving a Kre\uın-like formula where the role of the deficiency spaces is played by the dual pair $(\X,\X')$; the parameter $Θ$ belongs to the space of symmetric operators from $\X'$ to $\X$. When $\X=\C$ one recovers the ``$\calH_{-2}$ -construction'' of Kiselev and Simon and so, to some extent, our results can be regarded as an extension of it to the infinite rank case. Considering the situation in which $\calH=L^2(\RE^n)$ and $τ$ is the trace (restriction) operator along some null subset, we give various applications to singular perturbations of non necessarily elliptic pseudo-differential operators, thus unifying and extending previously known results.
Proposition 2.1 revised. Remarks 2.15 and 2.16 added. 38 pages. To appear in Journal of Functional Analysis
Proposition 2.1 revised. Remarks 2.15 and 2.16 added. 38 pages. To appear in Journal of Functional Analysis