# Khodr Shamseddine

## Characterization Of Compact and Self-Adjoint Operators and Study Of Positive Operators On A Banach Space Over A Non-Archimedean Field

Date Friday, March 21, 2014

Let $c_0$ denote the space of all null sequences of the complex Levi-Civita field ${mathcal C}$. We define a natural inner product on $c_0$ which induces the sup-norm of $c_0$. Unlike classical Hilbert spaces, $c_0$ is not orthomodular with respect to this inner product, so we characterize those closed subspaces of $c_0$ that have orthonormal complements. We also present characterizations of normal projections, adjoint and self-adjoint operators, and compact operators on $c_0$. Then we work on some $B^*$-algebras of operators on $c_0$, including those mentioned above; and we define an inner product on such algebras that induces the usual norm of operators. Finally, we study the properties of positive operators, which we then use to introduce a partial order on the set of compact and self- adjoint operators on $c_0$ and study the properties of that partial order.