Compact perturbations of operators with property (t)

Abstract

Let ℋ{\mathcal{ {\mathcal H} }} be an infinite dimensional complex Hilbert space and ℬ(ℋ){\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) the algebra of all bounded linear operators on ℋ{\mathcal{ {\mathcal H} }}. For an operator T∈ℬ(ℋ)T\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}), we say property (t)\left(t) holds for TT if σ(T)⧹σuw(T)=Π00(T)\sigma \left(T)\hspace{-0.08em}\setminus \hspace{-0.08em}{\sigma }_{uw}\left(T)={\Pi }_{00}\left(T), where σ(T)\sigma \left(T) and σuw(T){\sigma }_{uw}\left(T) denote the spectrum and the Weyl essential approximate point spectrum of TT, respectively, and Π00(T)={λ∈isoσ(T):0<n(T−λ)<∞}{\Pi }_{00}\left(T)=\left\{\lambda \in {\rm{iso}}\sigma \left(T):0\lt n\left(T-\lambda )\lt \infty \right\}. In this paper, we consider the stability of property (t)\left(t) under (small) compact perturbations. Also, we explore the relations between the stability of property (t)\left(t) and the stability of Weyl-type theorems. Moreover, we characterize those operators TT satisfying that property (t)\left(t) holds for f(T)f\left(T) for each function ff analytic on some neighborhood of σ(T)\sigma \left(T).

Keywords

• property (t)
• compact perturbation
• weyl-type theorem
• function of operator
• 47b20
• 47a10