  Alfv\'en-like operators are of interest in magnetohydrodynamics, which is
used in plasma physics to study the macroscopic behavior of plasma. Motivated
by this important and complex application, we focus on a parameter-dependent
curl-div problem that can be seen as a prototype of an Alfv\'en-like operator,
and we discretize it using isogeometric analysis based on tensor-product
B-splines. The involved coefficient matrices can be very ill-conditioned, so
that standard numerical solution methods perform quite poorly here. In order to
overcome the difficulties caused by such ill-conditioning, a two-step strategy
is proposed. First, we conduct a detailed spectral study of the coefficient
matrices, highlighting the critical dependence on the different physical and
approximation parameters. Second, we exploit such spectral information to
design fast iterative solvers for the corresponding linear systems. For the
first goal we apply the theory of (multilevel block) Toeplitz and generalized
locally Toeplitz sequences, while for the second we use a combination of
multigrid techniques and preconditioned Krylov solvers. Several numerical tests
are provided both for the study of the spectral problem and for the solution of
the corresponding linear systems.
