Linear operator equations

Abstract

where / is holomorphic on a(u, Ct) — a(v, Ct), and c is a suitable contour. In the present note we study the operator SG® defined by Sx = 'YJi=iuixvi, {ui} and {vj} being commutative subsets of Ct; but it is understood that {uj} need not commute with {vj}. In a generic way, we shall refer to such an operator as an elementary operator.2 Of course this is related to the problem of solving in Ct a general system of linear equations. We generalize the analysis made in [4] covering the ux — xv case, and extend the following unpublished theorem of D. C. Kleinecke: (1.3) If Ct is the Banach algebra of all operators on a Banach space, then C may be replaced by = in (1.1).