  We consider an inverse spectral problem for a class of non-compact Hankel
operators $H$ such that the modulus of $H$ (restricted onto the orthogonal
complement to its kernel) has simple spectrum. Similarly to the case of compact
operators, we prove a uniqueness result, i.e. we prove that a Hankel operator
from our class is uniquely determined by the spectral data. In other words, the
spectral map, which maps a Hankel operator to the spectral data, is injective.
Further, in contrast to the compact case, we prove the failure of surjectivity
of the spectral map, i.e. we prove that not all spectral data from a certain
natural set correspond to Hankel operators. We make some progress in describing
the image of the spectral map. We also give applications to the cubic Szeg\H{o}
equation. In particular, we prove that not all solutions with initial data in
BMOA are almost periodic; this is in a sharp contrast to the known result for
initial data in VMOA.
