Limiting absorption principle for perturbed operator

TL;DR

This paper proves a theorem on the limiting absorption principle for perturbed self-adjoint operators in Hilbert spaces, establishing conditions under which the resolvent operators have norm limits as the spectral parameter approaches the real axis.

Contribution

It introduces a new result on the limiting absorption principle for perturbed operators, extending previous work to include invariant operator ideals.

Findings

The resolvent of the perturbed operator has a norm limit under specified conditions.

The result applies to a broad class of operators including invariant operator ideals.

Provides a framework for analyzing spectral properties of perturbed operators.

Abstract

In this note the following theorem is proved. Let $\mathcal H$ and $\mathcal K$ be Hilbert spaces. Let $H_0$ be a self-adjoint operator on $\mathcal H,$ $F \colon \mathcal H \to \mathcal K$ be a closed $|H_0|^{1/2}$-compact operator, and $J \colon \mathcal K \to \mathcal K$ be a bounded self-adjoint operator. If the operator $$ F (H_0 - \lambda - iy)^{-1} F^* $$ has norm limit as $y \to 0^+$ for a.e.~$\lambda,$ then so does the operator $$ F (H_0 + F^*JF - \lambda - iy)^{-1} F^*. $$ An invariant operator ideal version of this result is also discussed.