Weak Compactness and Weak Essential Spectra of Elementary Operators

Let E be a Banach space and let A = (A1,...,An) and B = (B1,...,Bn) be n-tuples of operators on E. The elementary operator ℰA,B : L(E) → L(E) is defined by $\mathcal{E}_{A,B} = \textstyle\sum ^{n}_{i=1} L_{A_{i}}R_{B_{i}}$, where LT and RT denote the multiplication operators LTU = TU and RTU = UT for U ∈ L(E). This paper studies weak compactness and spectra modulo W(L(E)) of elementary operators, and it extends the most important results obtained previously for single two-sided multiplication operators. In the case that E = ℓp, 1 < p < ∞, we give a characterization for the weak compactness of ℰA,B and show that σw(ℰA,B) = σTe(A) ο σTe(B), where σTe denotes the Taylor essential spectrum and A and B are commuting n-tuples of operators. Similarly, in the case that E′ has the Dunford-Pettis property we characterize the weak compactness and show that σw(ℰA,B) = σe(ℰA,B) = σTe(A) ο σT(B) ∪ σT(A) ο σTe(B), where σT denotes the Taylor spectrum. The essential spectrum of ℰA,B has been computed before by Curto, Fialkow and Eschmeier, and our techniques yield also a new proof of their result. Most of our results remain valid for restrictions of ℰA,B to Banach ideals I of L(E). A weakly compact analogue of the Stampfli identity for the norm of an inner derivation is established by computing the distance to W(L(ℓ2)) of an inner derivation on L(ℓ2). Finally, we determine the essential norm of a two-sided multiplication operator on L(ℓ2) or on ideals I of L(ℓ2).