General explicit descriptions for intertwining operators and direct rotations of two orthogonal projections

Dou, Yan-Ni; Shi, Weijuan; Miaomiao, Cui; Du, Hongliang

doi:10.1016/j.laa.2017.06.036

Cited by 15 publications

(11 citation statements)

References 17 publications

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“…Similarly, P |M 10 = 0, Q|M 10 = I. This agrees with (2) and allows us to consider now the restrictions of A, B to the orthogonal complement H ′ of M 01 ⊕ M 10 .…”

Section: The Supersymmetric Approachsupporting

confidence: 80%

“…so that (26) holds in a trivial way. A description of all U satisfying (25) was provided in [10]. In the notation (2) it looks as follows [4]:…”

Section: Intertwiningmentioning

confidence: 99%

“…The approach of [1,9,10] and [24] was geometrical, using either (2) or its equivalents. In [17] a point was made to derive the existence criterion for the intertwining unitary U via the supersymmetric approach.…”

Section: The Supersymmetric Approachmentioning

confidence: 99%

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Robert Sheckley’s Answerer for two orthogonal projections

Böttcher

Spitkovsky

2018

The Diversity and Beauty of Applied Operator Theory

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This paper is the written version of our talk (presented by the second author) at the IWOTA in Chemnitz in August 2017. The meta theorem of the paper is that Halmos' two projections theorem is something like Robert Sheckley's Answerer: no question about the W * -and C * -algebras generated by two orthogonal projections will go unanswered, provided the question is not foolish. An alternative approach to questions about two orthogonal projections makes use of the supersymmetry equality introduced by Avron, Seiler, and Simon. A noteworthy insight of the paper reveals that the supersymmetric approach is nothing but Halmos in different language and hence an equivalent Answerer.

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“…Similarly, P |M 10 = 0, Q|M 10 = I. This agrees with (2) and allows us to consider now the restrictions of A, B to the orthogonal complement H ′ of M 01 ⊕ M 10 .…”

Section: The Supersymmetric Approachsupporting

confidence: 80%

“…so that (26) holds in a trivial way. A description of all U satisfying (25) was provided in [10]. In the notation (2) it looks as follows [4]:…”

Section: Intertwiningmentioning

confidence: 99%

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Robert Sheckley’s Answerer for two orthogonal projections

Böttcher

Spitkovsky

2018

The Diversity and Beauty of Applied Operator Theory

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“…where P 1 ∈ B(R(P ) ⊥ , R(P )). Also, we denote by In recent years, the descriptions for intertwining operators and a fixed difference properties of two orthogonal projections are considered in [4,7,15,16,17]. That is how to find a unitary operator U such that U P = QU and U Q = P U for projections P and Q.…”

Section: Introductionmentioning

confidence: 99%

The structures and decompositions of symmetries involving idempotents

Zhang

Wei

2020

Banach J. Math. Anal.

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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       

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“…Finally, in case elements V with the required properties exist, we want to describe all of them. As we were polishing our paper, we learned that [6] had addressed the same problem with some but certainly not complete overlap as we will explain below. We begin with a simple observation.…”

Section: Introductionmentioning

confidence: 99%

Similarity Between Two Projections

Böttcher

Simon

Spitkovsky

2017

Integr. Equ. Oper. Theory

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Abstract. Given two orthogonal projections P and Q, we are interested in all unitary operators U such that U P = QU and U Q = P U . Such unitaries U have previously been constructed by Wang, Du, and Dou and also by one of the authors. One purpose of this note is to compare these constructions. Very recently, Dou, Shi, Cui, and Du described all unitaries U with the required property. Their proof is via the two projections theorem by Halmos. We here give a proof based on the supersymmetric approach by Avron, Seiler, and one of the authors.Mathematics Subject Classification (2010). Primary 47A62, Secondary 46H15, 46L89, 47A67, 47C15.

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General explicit descriptions for intertwining operators and direct rotations of two orthogonal projections

Cited by 15 publications

References 17 publications

Robert Sheckley’s Answerer for two orthogonal projections

Robert Sheckley’s Answerer for two orthogonal projections

The structures and decompositions of symmetries involving idempotents

Similarity Between Two Projections

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