Idempotents as J-Projections

Matvejchuk, Marjan

doi:10.1007/s10773-011-0772-4

Cited by 12 publications

(12 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By JpJ, p * ∈ B(H), it will suffice to show that Jpx = JpJx = p * x for all x ∈ H . It is easily to see this fact from (9), and equalities (10), (11). Thus JpJ = p * .…”

mentioning

confidence: 76%

“…Note that the indefinite analog of Theorem 2 was proved in [8] (see also [9]). Theorem 2 was announced without proof in [10].…”

mentioning

confidence: 88%

“…Put Jx = x − ix for all x ∈ H, x = x + ix , where x , x ∈ H . By (9), and by the construction of H , the operators X, (X 2 − X) 1/2 , V , (X − Q + ) are J-real operators. By (3), (4), (8), an easy computation shows that…”

mentioning

confidence: 99%

See 2 more Smart Citations

Idempotents as J-projections: II

Matvejchuk

2012

Lobachevskii J Math

View full text Add to dashboard Cite

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.

show abstract

“…By JpJ, p * ∈ B(H), it will suffice to show that Jpx = JpJx = p * x for all x ∈ H . It is easily to see this fact from (9), and equalities (10), (11). Thus JpJ = p * .…”

mentioning

confidence: 76%

“…Note that the indefinite analog of Theorem 2 was proved in [8] (see also [9]). Theorem 2 was announced without proof in [10].…”

mentioning

confidence: 88%

See 1 more Smart Citation

Idempotents as J-projections: II

Matvejchuk

2012

Lobachevskii J Math

View full text Add to dashboard Cite

show abstract

“…Furthermore, some decomposition properties of projections (or J-projections) were studied in [2,6,11]. In particular, the existence of J-selfadjoint (positive, contractive) projections and its properties are obtained in [12,13,14]. Also, the minimal and maximal elements of the set of all symmetries the symmetries J with P * JP J (or JP 0) are given in [9,10].…”

Section: Introductionmentioning

confidence: 99%

The structures and decompositions of symmetries involving idempotents

Zhang

Wei

2020

Banach J. Math. Anal.

View full text Add to dashboard Cite

Xi ′ an, 710062, P eople ′ s Republic of China. b. College of Xingzhi, Xi ′ an U niversity of F inance and Economics, Xi ′ an, 710038, P eople ′ s Republic of China.Abstract Let H be a separable Hilbert space and P be an idempotent on H. We denote byIn this paper, we first get that symmetries (2P − I)|2P − I| −1 and (P + P * − I)|P + P * − I| −1 are the same. Then we show that Γ P = ∅ if and only if ∆ P = ∅. Also, the specific structures of all symmetries J ∈ Γ P and J ∈ ∆ P are established, respectively. Moreover, we prove that J ∈ ∆ P if and only ifwhere U is a unitary operator from R(P ) ⊥ onto R(P ) with U * P 1 = P * 1 U. Proof. Suppose that J has the following operator matrix form J = J 11 J 12 J * 12 J 22 : R(P ) ⊕ R(P ) ⊥ , where J 11 and J 22 are self-adjoint operators. It follows from the fact JP = (I − P )J that        J 11 = −P 1 J * 12 1 J 11 P 1 = −P 1 J 22 2 J * 12 P 1 = J 22 3 . (2.9) 6 On the other hand, J = J * = J −1 yields J 2 = I, so       

show abstract

“…In recent years, the existence of J-positive (negative, contractive, expansive) projections and its properties are considered in [6,7,8]. And some geometry and topological properties of projections and decomposition properties of J−projections were studied in [3,5,[9][10][11][12][13]. In particular, an exposition of operators in Krein spaces can be found in the lecture by T. Ando [1].…”

Section: Introductionmentioning

confidence: 99%

The minimal and maximal symmetries for J-contractive projections

Cai

Niu

et al. 2019

Linear Algebra and its Applications

View full text Add to dashboard Cite

In this note, we firstly consider the structures of symmetries J such that a projection P is J-contractive. Then the minimal and maximal elements of the symmetries J with P * JP J(or JP 0) are given. Moreover, some formulas between P (2I−P −P * ) + (P (2I−P −P * ) − ) and P (P +P * ) − (P (P +P * ) + ) are established.: 4)where J 1 ∈ B(R(P )) is a symmetry with J 1 P 1 + P 1 = 0. Proof. Necessity. We assumewhere J 11 and J 22 are self-adjoint operators. The equation of J = J * = J −1 implies that  .(2.6)Using equations 1 of (2.5) and (2.6), we have J 11 (I + P 1 P * 1 )J 11 = I,

show abstract

Idempotents as J-Projections

Cited by 12 publications

References 4 publications

Idempotents as J-projections: II

Idempotents as J-projections: II

The structures and decompositions of symmetries involving idempotents

The minimal and maximal symmetries for J-contractive projections

Contact Info

Product

Resources

About