WEAK COMPACTNESS AND WEAK ESSENTIAL SPECTRA OF ELEMENTARY OPERATORS

Let E be a Banach space and let A = (A(1),..., A(n)) and B = (B-1,..., B-n) be n-tuples of operators on E. The elementary operator E (A,B) : L(E) --> L(E) is defined by E(A,B) = Sigma(i=1)(n) L(Ai) R(Bi), where L(T) and R(T) denote the multiplication pn operators L(T)U = TU and R( T)U = UT for U is an element of L(E). This paper studies weak compactn ess and spectra module W(L(E)) of elementary operators, and it extends the most important results obtained previously for single two-sided m ultiplication operators. In the case that E = l(P), 1 < p < infinity, we give a characterization for the weak compactness of E(A,B) and show that sigma omega(E (A,B)) = sigma(Te)(A) o sigma(Te)(B), where sigma( e) denotes the Taylor essential spectrum and A and B are commuting n-t uples of operators. Similarly, in the case that E' has the Dunford-Pet tis property we characterize the weak compactness and show that sigma omega(E(A,B)) = sigma e,(E(A,B)) = sigma(Te)(A) o sigma(T)(B) boolean OR sigma(T)(A) o sigma(Te)(B), where sigma(T) denotes the Taylor spect rum. The essential spectrum of E(A,B) has been computed before by Curt o, Fialkow and Eschmeier, and our techniques yield also a new proof of their result. Most of our results remain valid for restrictions of E( A,B) to Banach ideals I of L(E). A weakly compact analogue of the Stam pfli identity for the norm of an inner derivation is established by co mputing the distance to W(L(l(2))) of an inner derivation on L(l(2)). Fi nally, we determine the essential norm of a two-sided multiplicatio n operator on L(l(2)) or on ideals I of L(l(2)).