**Salah Mecheri**

## Weyl's Theorem for Algebraically (*p*,*k*)-Quasihyponormal
Operators

**Abstract:**

Let $A$ be a bounded linear operator acting on a Hilbert space $H$. The $B$-Weyl
spectrum of $A$ is the set $\sigma_{Bw}(A)$ of all $\lambda\in\mathbb C$ such
that $A-\lambda I$ is not a $B$-Fredholm operator of index 0. Let $E(A)$ be the
set of all isolated eigenvalues of $A$. In 2004 M. Berkani and A. Arroud showed
that if $A$ is hyponormal, then $A$ satisfies the generalized Weyl's theorem $\sigma_{Bw}(A)=\sigma(A)\setminus
E(A)$, and the $B$-Weyl spectrum $\sigma_{Bw}(A)$ of $A$ satisfies the spectral
mapping theorem. Lee in 2000 showed that Weyl's theorem holds for algebraically
hyponormal operators. In this paper the above results are generalized to an
algebraically ($p,k$)-quasihyponormal operator which includes an algebraically
hyponormal operator.

**Keywords:**

Hyponormal operator, $(p,k)$-quasihyponormal operator, Generalized Weyl's
theorem, Browder's theorem.

**MSC 2000:** 47A10, 47A12, 47B20