Similarity Between Two Projections

Böttcher, Albrecht; Simon, Barry; Spitkovsky, Ilya M.

doi:10.1007/s00020-017-2414-6

Cited by 14 publications

(3 citation statements)

References 19 publications

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“…However, with contemporary computational resources, our algorithm (and Trench's as a special instance) becomes extremely effective: it combines, in a way that is optimal for the task, physical, mathematical, and procedural insight. Let us also mention in passing that a problem that has been much investigated in the non-block case is associated to Toeplitz matrices perturbed by impurities [21]. Our work can easily extend this investigations to the block case, where there is considerable room for surprises from a physical perspective [22].…”

Section: Introductionmentioning

confidence: 71%

Exact solution of corner-modified banded block-Toeplitz eigensystems

Cobanera¹,

Alase²,

Ortíz³

et al. 2017

J. Phys. A: Math. Theor.

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Abstract. Motivated by the challenge of seeking a rigorous foundation for the bulkboundary correspondence for free fermions, we introduce an algorithm for determining exactly the spectrum and a generalized-eigenvector basis of a class of banded block quasi-Toeplitz matrices that we call corner-modified. Corner modifications of otherwise arbitrary banded block-Toeplitz matrices capture the effect of boundary conditions and the associated breakdown of translational invariance. Our algorithm leverages the interplay between a non-standard, projector-based method of kernel determination (physically, a bulk-boundary separation) and families of linear representations of the algebra of matrix Laurent polynomials. Thanks to the fact that these representations act on infinite-dimensional carrier spaces in which translation symmetry is restored, it becomes possible to determine the eigensystem of an auxiliary projected block-Laurent matrix. This results in an analytic eigenvector Ansatz, independent of the system size, which we prove is guaranteed to contain the full solution of the original finitedimensional problem. The actual solution is then obtained by imposing compatibility with a boundary matrix, also independent of system size. As an application, we show analytically that eigenvectors of short-ranged fermionic tight-binding models may display power-law corrections to exponential decay, and demonstrate the phenomenon for the paradigmatic Majorana chain of Kitaev.

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Section: Introductionmentioning

confidence: 71%

Exact solution of corner-modified banded block-Toeplitz eigensystems

Cobanera¹,

Alase²,

Ortíz³

et al. 2017

J. Phys. A: Math. Theor.

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show abstract

“…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%

“…That is how to find a unitary operator U such that U P = QU and U Q = P U for projections P and Q. For a pair (P, Q) of orthogonal projections, the characterization of intertwining operator U is given in [4,7,14,17]. Indeed, these results also describe the sufficient and necessary condition for the existence of a symmetry J with JP J = Q and the explicit formulas of all symmetries J with JP J = 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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Operator Projective Line and Its Transformations

Aljasem,

Kisil

2024

Analysis Without Borders

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Similarity Between Two Projections

Cited by 14 publications

References 19 publications

Exact solution of corner-modified banded block-Toeplitz eigensystems

Exact solution of corner-modified banded block-Toeplitz eigensystems

The structures and decompositions of symmetries involving idempotents

Operator Projective Line and Its Transformations

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