Generalized Browder's and Weyl's theorems for left and right multiplication operators (Q984802)

For a bounded linear operator \(T\) on a complex Banach space \(X\), let \(\sigma(T)\) be the spectrum of \(T\) and \(E_{0}(T)\subseteq \sigma(T)\) the set of all isolated eigenvalues of finite multiplicity. A Fredholm operator of index zero is said to be a Weyl operator and the Weyl spectrum of \(T\) is the set \(\sigma_{W}(T)\) of all those \(\lambda \in \sigma(T)\) such that \(T-\lambda\) is not a Weyl operator. If \(\sigma_{W}(T)=\sigma(T)\setminus E_{0}(T)\), then it is said that Weyl's theorem holds for \(T\). There are several generalizations and analogues of this definition: generalized Weyl's theorem, \(a\)-Weyl's theorem, Browder's theorem, etc. Let \(A\) and \(B\) be bounded linear operators on \(X\). Then one defines multiplication operators on the Banach algebra \(B(X)\) of all bounded linear operators on \(X\) as follows: \(L_AT=AT\), \(R_BT=TB\), and \(\tau_{AB}T=ATB\), where \(T\in B(X)\) is arbitrary. The main objective of the paper is the study of Weyl's theorem and its generalizations for the mentioned multiplication operators. For instance, for every \(A\in B(X)\), one has \(E_{0}(L_A)=E_{0}(R_A)=\emptyset\), which means that Weyl's theorem holds for \(L_A\) and \(R_A\). Moreover, it is shown that \(a\)-Weyl's theorem holds for \(L_A\) and \(R_A\). There are several results similar to the following Theorem. The generalized Weyl theorem holds for \(A\) if and only if Browder's theorem holds for \(A\) and generalized Weyl's theorem holds for \(L_A\). The paper is well written and includes many interesting results.