  We introduce a certain differential (heat) operator on the space of Hermitian
Jacobi forms of degree 1, show it's commutation with certain Hecke operators
and use it to construct a lift of elliptic cusp forms to Hermitian Jacobi cusp
forms. We construct Hermitian Jacobi forms as the image of the tensor product
of two copies of Jacobi forms and also from differentiation of the variables.
We determine the number of Fourier coefficients that determine a Hermitian
Jacobi form and use it to embed a certain subspace of Hermitian Jacobi forms
into a direct sum of modular forms for the full modular group.
