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

摘要

Let C be the complex Levi-Civita field and let E be a free Banach space over C of countable type. Then E is isometrically isomorphic to c_0,(N, C, s):= {(x+n)_n εN: x_n ε C; limn_n→∞|x_n|s(n) = 0} where s: N → (0,∞) If the range of s is contained in |C {0}| it is enough to study c_0 (N, C),, which will be denoted by c_0(C) or, simply, c_0. In this paper, we define a natural inner product on c_0, which induces the sup-norm of c_0. Of course, c_0 is not orthomodular, so we characterize those closed subspaces of c_0 with an orthonormal complement with respect to this inner product; that is, those closed subspaces M of c_0 such that c_0 = 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 c_0.