Compact operators on non-classical Hilbert spaces

For a class of Norm Hilbert spaces E over a Krull valued field K (including many Form Hilbert spaces) the Banach algebra Lip(E) of all linear Lipschitz operators E --> E is studied. A notion of supercompact operator is introduced leading to a closed two-sided ideal SC(E) of Lip(E). Many striking differences with the classical theory will be established, such as the commutativity of the Calkin algebra Lip(E)/SC(E) and the fact that each supercompact operator has a trace. If K is not spherically complete we will prove that SC(E) and Lip(E) are in a natural way each other's dual, showing that both Banach spaces are reflexive.