  In this work we focus on the preconditioning of a Galerkin space-time
isogeometric discretization of the heat equation. Exploiting the tensor product
structure of the basis functions in the parametric domain, we propose a
preconditioner that is the sum of Kronecker products of matrices and that can
be efficiently applied thanks to an extension of the classical Fast
Diagonalization method. The preconditioner is robust w.r.t. the polynomial
degree of the spline space and the time required for the application is almost
proportional to the number of degrees-of-freedom, for a serial execution. By
incorporating some information on the geometry parametrization and on the
equation coefficients, we keep high efficiency with non-trivial domains and
variable thermal conductivity and heat capacity coefficients.
