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 of the complexification of a real Banach space B,if commutes with complex conjugation, and if is compact for some , then if N is the number of points in (mod 2). Under further assumptions on and y (see Theorem 10 below), deg and deg (mod 2). Our results generalize some recent work of Jane Cronin [6][7].