Idempotents as J-projections: II

Abstract

Let B(H) Id be the set of all bounded idempotents on a complex Hilbert space H and let J be a conjugation operator on H. Fix p ∈ B(H) Id . At the paper we describe of J-projections. We prove that for a given p there exists a conjugation operator J 0 such that p is a J 0 -projection. In ([1], Chapter XII) the problem of construction of probability theory for quantum mechanics is posed. An analog of boolean algebra of events is quantum logic. An important interpretation of a quantum logic is the set B(H) or of all orthogonal (=self-adjoint) projections on a Hilbert space H. The problem to construct a quantum field theory sometimes leads to an indefinite metric space. In the indefinite case, the set B(H) Jo of all J-orthogonal projections is an analog of the logics B(H) or . It is important to know the properties and structure of projections in construction of measure theory on projections logic. J-projections were extensively studied at [2-6]. Let H be a complex Hilbert space with the Hilbert product (·, ·). Let B(H) be the set of all bounded operators on H. Let J be a conjugation operator in H (see [7] $50), i.e. 1) J 2 = I, 2) (Jx, Jy) = (y, x), for all x, y ∈ H. Note by 1), and 2), J(λx + βy) = λJx + βJy, for all λ, β ∈ C. A vector x ∈ H is said to be J-real if Jx = x. The vectors x := 1 2 (x + Jx) and x := 1 2i (x − Jx) = − 1 2 (ix + Jix) are J-real,∀x ∈ H and x = x + ix . The set H of all J-real vectors is a real Hilbert space with respect to the inner product (·, ·). Put x, y := (Jx, y). It is clear that an operator A ∈ B(H) is J-self-adjoint, i.e. Ax, y = x, Ay , ∀x, y ∈ H iff A = JA * J. Any bounded J-self-adjoint idempotent (=projection) p is said to be J-projection.Ap is a J 0 -projection (i.e. p = J 0 p * J 0 ) with respect to the symmetry J 0 := 2p − I and a J 1 -projection with respect to any conjugation operator J 1 such that J 1 pH = pH, J 1 (I − p)H = (I − p)H.We can formulate our main result Theorem 2. Let p be a bounded idempotent on a complex Hilbert space H. Then there exists a conjugation operator J on H with (·, ·) such that p is the J-projection.