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Abstract. We consider linear operators defined on a subspace of a complex Banach space into its topological antidual acting positively in a natural sense. The goal of this paper is to investigate of this kind of operators. The main theorem is a constructive characterization of the bounded positive extendibility of these linear mappings. From this result we can characterize the compactness of the extended operators and that when the positive extensions have closed ranges.As a main application of our general extension theorem, we present some necessary and sufficient conditions that a positive functional defined on a left ideal of a Banach * -algebra admits a representable positive extension. The approach we use here is completely constructive.

T * T is selfadjoint if T is a densely defined closed Hilbert space operator. This result of von Neumann can be generalized for not necessarily closed operators: T * T always admits a positive selfadjoint extension. The Friedrichs extension also will be obtained whenever T * T is assumed to be densely defined. Selfadjointness of T * T will be investigated. Densely defined positive operators and their Friedrichs extension A and AF , respectively, will be described by showing the existence of a closable operator T such that A = T

The purpose of this paper is to revise von Neumann’s characterizations of selfadjoint operators among symmetric ones. In fact, we do not assume that the underlying Hilbert space is complex, nor that the corresponding operator is densely defined, moreover, that it is closed. Following Arens, we employ algebraic arguments instead of the geometric approach of von Neumann using the Cayley transform.

Abstract. We provide sufficient and necessary conditions guaranteeing equations (A + B) * = A * + B * and (AB) * = B * A * concerning densely defined unbounded operators A, B between Hilbert spaces. We also improve the perturbation theory of selfadjoint and essentially selfadjoint operators due to Nelson, Kato, Rellich, and Wüst. Our method involves the range of two-by-two matrices of the form M S,T = I −T S I that makes it possible to treat real and complex Hilbert spaces jointly.