  Let $L_0$ be a bounded operator on a Banach space, and consider a
perturbation $L=L_0+K$, where $K$ is compact. This work is concerned with
obtaining bounds on the number of eigenvalues of $L$ in subsets of the
complement of the essential spectrum of $L_0$, in terms of the approximation
numbers of the perturbing operator $K$. Our results can be considered as wide
generalizations of classical results on the distribution of eigenvalues of
compact operators, which correspond to the case $L_0=0$. They also extend
previous results on operators in Hilbert space. Our method employs complex
analysis and a new finite-dimensional reduction, allowing us to avoid using the
existing theory of determinants in Banach spaces, which would require strong
restrictions on $K$. Several open questions regarding the sharpness of our
results are raised, and an example is constructed showing that there are some
essential differences in the possible distribution of eigenvalues of operators
in general Banach spaces, compared to the Hilbert space case.
