Involutions on surfaces with pg = q=1

In this paper some numerical restrictions for surfaces with in involution are obtained. These formulas are used to study surfaces of general type S with p(g) = q = 1 having an involution i such that S/i is a non-ruled surface and such that the bicanonical map of S is not composed with i. A complete list of possibilities is given and several new examples are constructed, as bidouble covers of surfaces. In particular the first example of a minimal surface of general type with p(g) = q = 1 and K-2 = 7 having birational bicanonical map is obtained.