  In this paper we introduce a new subspace of Jacobi forms of higher degree
via certain relations among Fourier coefficients. We prove that this space can
also be characterized by duality properties of certain distinguished embedded
Hecke operators. We then show that this space is Hecke invariant with respect
to all good Hecke operators.
  As explicit examples we give Eisenstein series.
  Conversely we show the existence of forms that are not contained in this
space.
