Compact operators in non-classical Hilbert spaces

H. Ochsenius
Department of Mathematics
Pontifice Universidad Católica de Chile
Casilla 306 - Correo 22
Santiago, Chile

W.H. Schikhof
Department of Mathematics
University of Nijmegen, Toernooiveld
6525 ED Nijmegen, The Netherlands

Date:

We consider Banach spaces E over a complete valued field K with a surjective infinite rank valuation $\vert \quad \vert : K \to G \cup \{ 0 \}$. Here G is a totally ordered abelian group which is the union of a strictly increasing sequence of convex subgroups, augmented with a smallest element 0. We assume that the values of the norm $\Vert \quad \Vert$ on E lie in $\Gamma \cup \{ 0 \}$, where $\Gamma$ is a totally ordered group containing G as a cofinal subgroup. Furthermore we require that E has a normorthogonal base $e_1,e_2,\ldots$ and that the sets $G \Vert e_n \Vert$ `are moving further and further away' from $\{ 1 \}$. It follows that E is a so-called norm Hilbert space (i.e. each closed subspace admits a continuous linear surjective projection $P : E \to D$ such that $\Vert Px \Vert \le \Vert x \Vert$ for all $x \in E$).
Examples will be described, including inner product spaces: it is shown that $\Vert \quad \Vert$ comes from an inner product iff $\Gamma = \sqrt G$.
Next, the Banach algebra Lip(E) of all linear Lipschitz operators $E \to E$ is studied. A notion of supercompactness is introduced leading to the closed two-sided ideal SC(E) in Lip(E). It will be shown that the Calkin algebra Lip(E)/SC(E) is commutative, but not a field, that all (semi-) Fredholm operators have zero index. By using the trace function on SC(E) a duality between SC(E) and Lip(E) will be established.