  This paper deals with the generalized spectrum of continuously invertible
linear operators defined on infinite dimensional Hilbert spaces. More
precisely, we consider two bounded, coercive, and self-adjoint operators
$\bc{A, B}: V\mapsto V^{\#}$, where $V^{\#}$ denotes the dual of $V$, and
investigate the conditions under which the whole spectrum of
$\bc{B}^{-1}\bc{A}:V\mapsto V$ can be approximated to an arbitrary accuracy by
the eigenvalues of the finite dimensional discretization
$\bc{B}_n^{-1}\bc{A}_n$. Since $\bc{B}^{-1}\bc{A}$ is continuously invertible,
such an investigation cannot use the concept of uniform (normwise) convergence,
and it relies instead on the pointwise (strong) convergence of
$\bc{B}_n^{-1}\bc{A}_n$ to $\bc{B}^{-1}\bc{A}$.
  The paper is motivated by operator preconditioning which is employed in the
numerical solution of boundary value problems. In this context, $\bc{A},
\bc{B}: H_0^1(\Omega) \mapsto H^{-1}(\Omega)$ are the standard
integral/functional representations of the differential operators $ -\nabla
\cdot (k(x)\nabla u)$ and $-\nabla \cdot (g(x)\nabla u)$, respectively, and
$k(x)$ and $g(x)$ are scalar coefficient functions. The investigated question
differs from the eigenvalue problem studied in the numerical PDE literature
which is based on the approximation of the eigenvalues within the framework of
compact operators.
  This work follows the path started by the two recent papers published in
[SIAM J. Numer. Anal., 57 (2019), pp.~1369-1394 and 58 (2020), pp.~2193-2211]
and addresses one of the open questions formulated at the end of the second
paper.
