We prove that if x = y+μ[δ(ü):-1Δ(μ)y is the unique solution of the equation x-üTx = y in a Banach space, with δ(ü) the Fredholm divisor of the operator T, and Δ an entire function of the complex variable ü, then Δ satisfies an exponential growth estimate of the form exp(clü|P). The proof holds for all operators T belonging to a certain quasi-normed operator ideal A with the property that A(p)supports a continuous trace. This result was known to hold for operators belonging to the approximative kernel of A, and it was conjectured by H König to be true generally. As a corollary we state general conditions implying the density of the set of eigenvectors of an operator.