Constructive compact operators on a Hilbert space

Abstract

In this paper, we deal with compact operators on a Hilbert space, within the framework of Bishop's constructive mathematics. We characterize the compactness of a bounded linear mapping of a Hilbert space into C n , and prove the theorems: (1) Let A and B be compact operators on a Hilbert space H, let C be an operator on H and let α ϵ C . Then α A is compact, A + B is compact, A ∗ is compact, CA is compact and if C ∗ exists, then AC is compact; (2) An operator on a Hilbert space has an adjoint if and only if it is weakly compact.