Property \(\mathcal {P}\) and Compact Perturbations

Let \(\mathcal {H}\) $\mathcal{H}$ be a complex separable infinite dimensional Hilbert space. An operator T acting on \(\mathcal {H}\) $\mathcal{H}$ is said to have property \(\mathcal {P}\) $\mathcal{P}$ , if \(\sigma (T)=\sigma _p(T)\) $\sigma (T)={\sigma }_p(T)$ and \(\sigma (T^*)=\sigma _p(T^*)\) $\sigma (T^{\ast })={\sigma }_p(T^{\ast })$ . In this paper, we characterize those operators which have an arbitrarily small compact perturbation to satisfy property \(\mathcal {P}\) $\mathcal{P}$ . Also, we study the stability of property \(\mathcal {P}\) $\mathcal{P}$ under small compact perturbations.