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On the Browder’s theorem of an elementary operator. (English) Zbl 1318.47006
A bounded linear operator $$T$$ on a complex Banach space $$X$$ is said to be a Weyl operator if it is a Fredholm operator of index zero. The Weyl spectrum of $$T$$ is $$\sigma_w(T)=\{ \lambda \in {\mathbb C}: T-\lambda I~\text{is not Weyl}\}$$. A Fredholm operator is said to be a Browder operator if its ascent and descent are finite; $$\sigma_b(T)=\{ \lambda \in {\mathbb C}: T-\lambda I \text{ is not Browder}\}$$ is the Browder spectrum of $$T$$. It is said that Browder’s theorem holds for $$T$$ if $$\sigma_w(T)=\sigma_b(T)$$.
Let $$A, B$$ be bounded linear operators on an infinite-dimensional complex Hilbert space $$H$$ and let $$d_{A,B}$$ denote either the elementary operator $$T\mapsto ATB-T$$ or $$T\mapsto AT-TB$$ on $$B(H)$$, the Banach algebra of all bounded linear operators on $$H$$. In this paper, some necessary and sufficient conditions on $$A$$ and $$B$$ are given such that Browder’s theorem holds for $$d_{A,B}$$.
##### MSC:
 47A10 Spectrum, resolvent 47A53 (Semi-) Fredholm operators; index theories 47B47 Commutators, derivations, elementary operators, etc.
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