Abstract:

Both Fa-Weyl’s theorem and a-Weyl’s theorem are the variants of Weyl’s theorem. The study of Weyl’s type theorems is very important for spectral theory. By defining a new spectral set in this paper, sufficient and necessary conditions for a bounded linear operator $T $ definded on a Hilbert space to satisfy the Fa-Weyl’s theorem and the a-Weyl’s theorem are established. In addition, we discuss the Fa-Weyl’s theorem and the a-Weyl’s theorem of bounded linear operator $T $ under a finite rank perturbation.

Key words: Fa-Weyl’s theorem, a-Weyl’s theorem, perturbation, spectrum