Transfer and entanglement stability of property ($UW${\normalsize\it{E}})

TL;DR

This paper investigates the stability and transfer of property (UW E) in operators and their functions, exploring spectral properties and entanglement stability in operator matrices.

Contribution

It introduces a new approach to analyze property (UW E) transfer via CI spectrum and studies entanglement stability in 2x2 operator matrices.

Findings

Property (UW E) transfer from T to f(T) and f(T*) established.

Spectral stability analyzed using CI spectrum and entanglement concepts.

Results applicable to upper triangular operator matrices.

Abstract

An operator $T\in B(H)$ is said to satisfy property ($UW${\scriptsize \it{E}}) if the complement in the approximate point spectrum of the essential approximate point spectrum coincides with the isolated eigenvalues of the spectrum. Via the CI spectrum induced by consistent invertibility property of operators, we explore property ($UW${\scriptsize \it{E}}) for $T$ and $T^\ast$ simultaneously. Furthermore, the transfer of property ($UW${\scriptsize \it{E}}) from $T$ to $f(T)$ and $f(T^{\ast})$ is obtained, where $f$ is a function which is analytic in a neighborhood of the spectrum of $T$. At last, with the help of the so-called $(A,B)$ entanglement stable spectra, the entanglement stability of property ($UW${\scriptsize \it{E}}) for $2\times 2$ upper triangular operator matrices is investigated.