Estimates for the Number of Solutions of Operator Equations

We define here a degree theory for proper, analytic Fredholm maps of index zero defined on open subsets of complex Banach spaces, and we prove that the standard properties for a degree theory hold. Our approach avoids the differential geometry tools used by Elworthy and Tromba [8] in similar work. We prove that for analytic maps the degree theory defined by Nussbaum in [11,13] agrees with ours, and similarly Browder and Gupta's degree theory in [3] is a special case of ours. If & is an analytic Fredholm map of index zero defined on an open subset g of the complexification 88 of a real Banach space B, if & commutes with complex conjugation, and if 8^~l(y) is compact for some y e F, then if N is the number of points in 10), N g deg (^-, g, y) and N e deg (tF, y) (mod 2). Under further assumptions on F = iFignB and y (see Theorem 10 below), deg (F, &r\\\\\\\\B, y) g N and deg (F, V (-1B, y) = N (mod 2). Our results generalize some recent work of Jane Cronin [6, 7].