Weyl's theorem for Algebraically class $A$ Operators

Abstract

Let $A$ be a bounded linear operator acting on a Hilbert space $H$. In [32], A. Uchiyama proved that Weyl's theorem holds for class A operators with the additional condition that $\ker A|_{[TH]}=0$ and he showed that every class A operator whose Weyl spectrum equals to zero is compact and normal. In this paper we show that Weyl's theorem holds for algebraically class $A$ operator without the additional condition $\ker A|_{[TH]}=0$. This leads as to show that a class $A$ operator whose Weyl spectrum equals to zero is always compact and normal.