  The decomposition or approximation of a linear operator on a matrix space as
a sum of Kronecker products plays an important role in matrix equations and
low-rank modeling. The approximation problem in Frobenius norm admits a
well-known solution via the singular value decomposition. However, the
approximation problem in spectral norm, which is more natural for linear
operators, is much more challenging. In particular, the Frobenius norm solution
can be far from optimal in spectral norm. We describe an alternating
optimization method based on semidefinite programming to obtain high-quality
approximations in spectral norm, and we present computational experiments to
illustrate the advantages of our approach.
