Characterization of compact and self-adjoint operators on free Banach spaces of countable type over the complex Levi-Civita field

Abstract

Let \documentclass[12pt]{minimal}\begin{document}$\mathcal {C}$\end{document}C be the complex Levi-Civita field and let E be a free Banach space over \documentclass[12pt]{minimal}\begin{document}$\mathcal {C}$\end{document}C of countable type. Then E is isometrically isomorphic to \documentclass[12pt]{minimal}\begin{document}$c_{0}\left( \mathbb {N},\mathcal {C},s\right)\break :=\left\lbrace (x_{n})_{n\in \mathbb {N}}:x_{n}\in \mathcal {C};\lim _{n\rightarrow \infty }|x_{n} |s(n)=0\right\rbrace$\end{document}c0N,C,s:=(xn)n∈N:xn∈C;limn→∞|xn|s(n)=0, where \documentclass[12pt]{minimal}\begin{document}$s:\mathbb {N}\rightarrow \left( 0,\infty \right) .$\end{document}s:N→0,∞. If the range of s is contained in \documentclass[12pt]{minimal}\begin{document}$\left|\mathcal {C}\setminus \left\lbrace 0\right\rbrace \right|,$\end{document}C∖0, it is enough to study \documentclass[12pt]{minimal}\begin{document}$c_{0}\left( \mathbb {N},\mathcal {C} \right)$\end{document}c0N,C, which will be denoted by \documentclass[12pt]{minimal}\begin{document}$c_{0}(\mathcal {C})$\end{document}c0(C) or, simply, c0. In this paper, we define a natural inner product on c0, which induces the sup-norm of c0. Of course, c0 is not orthomodular, so we characterize those closed subspaces of c0 with an orthonormal complement with respect to this inner product; that is, those closed subspaces M of c0 such that c0 = M ⊕ M⊥. Such a subspace, together with its orthonormal complement, defines a special kind of projection, the so-called normal projection. We present a characterization of such normal projections as well as a characterization of another kind of operators, the compact operators on c0.