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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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