Weakly ordered partial commutative group of self-adjoint linear operators densely defined on Hilbert space

Abstract

ABSTRACT. We continue in a direction of describing an algebraic structure of linear operators on infinite-dimensional complex Hilbert space H. In [Paseka, J.--Janda, J.: More on PT-symmetry in (generalized) effect algebras and partial groups, Acta Polytech. 51 (2011), 65-72] there is introduced the notion of a weakly ordered partial commutative group and showed that linear operators on H with restricted addition possess this structure. In our work, we are investigating the set of self-adjoint linear operators on H showing that with more restricted addition it also has the structure of a weakly ordered partial commutative group.

Weakly Ordered A-Commutative Partial Groups of Linear Operators Densely Defined on Hilbert Space

“…They also showed that the set of all linear operators on complex Hilbert space H with the usual sum, which is restricted to the same domain for unbounded operators (partial operation ⊕ D ), possesses this structure. In [4] we considered the structure on the important subset of self-adjoint operators, showing that it is also a wop-group.…”

“…Wop-groups have only a non-constructive associativity (the equation holds if and only if both sides are defined). It has been shown [4] that the set of all linear operators has generally stronger algebraic properties. This was a motivation for introducing the notion of a weakly ordered partial a-commutative group (woa-group) where the associative law is more constructive.…”

“…and operation ⊕ u by A ⊕ u B = A ⊕ D B iff at least one of A, B ∈ Gr(H) is bounded or A, B ∈ Gr(H) are both unbounded, D(A) = D(B) and there exists a real number λ A B = 0 such that A − λ A B B is bounded [4]. For an arbitrary subset X ⊆ Gr(H) let us define a relation…”