Abstract. A bounded linear operator T ∈ L(X) on a Banach space X is said to satisfy "Weyl's theorem" if the complement in the spectrum of the Weyl spectrum is the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity. In this paper we show that if T is a paranormal operator on a Hilbert space, then T + K satisfies Weyl's theorem for every algebraic operator K which commutes with T .
Preliminaries and notationLet L(X) be the Banach algebra of all bounded linear operators on an infinite- Obviously B(X) ⊆ W (X). The Weyl spectrum and the Browder spectrum of T ∈ L(X) are defined by σ w (T ) := {λ ∈ C : λI − T / ∈ W (X)}
We introduce the spectral property (R), for bounded linear operators defined on a Banach space, which is related to Weyl type theorems. This property is also studied in the framework of polaroid ,or left polaroid, operators.Mathematics Subject Classification (2010). Primary 47A10, 47A11. Secondary 47A53, 47A55.
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