Weak essential spectra of multiplication operators on spaces of bounded linear operators (Q1337555)

Let \(E\) be a (complex) Banach space and consider the multiplication operator \(A\land B: S\mapsto BSA\) on \(L(E)\) for \(A,B\in L(E)\). The spectrum of \(A\land B\in L(L(\ell^ p))\) modulo \(W(L(\ell^ p))\) is determined. In addition, a general spectral inclusion formula is established which enables one to compute the spectrum of \(A\land B\) modulo \(W(L(E))\) if \(E'\) has the Dunford-Pettis property. The distance of \(A\land B\) to the weakly compact operators on \(L(\ell^ p)\) is calculated. Similar results hold for the restriction of \(A\land B\) to Banach ideals \(I(E)\) of \(L(E)\). This study complements several previous results concerning the spectrum and the essential spectrum of \(A\land B\).