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

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.