Function theory and spectral mapping theorems for antilinear operators

Huhtanen, Marko; Perämäki, Allan

doi:10.7900/jot.2013may20.1991

Cited by 2 publications

(2 citation statements)

References 17 publications

(37 reference statements)

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“…The intersection of two ϑ-normal subspaces is a ϑ-normal subspace. If one of the two subspaces is minimal, then their intersection is either the minimal ϑ-normal subspace itself or it consists of the zero vector only, see also [43]. Hence: This assumption implies the uniqueness of the polar decomposition: There is a unique antiunitary operator θ such that ϑ = θ|ϑ| = |ϑ|θ.…”

Section: Decompositions Of Normal Antilinear Operatorsmentioning

confidence: 99%

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Anti- (conjugate) linearity

Uhlmann

2016

Sci. China Phys. Mech. Astron.

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This is an introduction to antilinear operators. In following E. P. Wigner the terminus "antilinear" is used as it is standard in Physics. Mathematicians prefer to say "conjugate linear".By restricting to finite-dimensional complex-linear spaces, the exposition becomes elementary in the functional analytic sense. Nevertheless it shows the amazing differences to the linear case.Basics of antilinearity is explained in sections 2, 3, 4, 7 and in subsection 1.2: Spectrum, canonical Hermitian form, antilinear rank one and two operators, the Hermitian adjoint, classification of antilinear normal operators, (skew) conjugations, involutions, and acq-lines, the antilinear counterparts of 1-parameter operator groups. Applications include the representation of the Lagrangian Grassmannian by conjugations, its covering by acq-lines. as well as results on equivalence relations. After remembering elementary Tomita-Takesaki theory, antilinear maps, associated to a vector of a two-partite quantum system, are defined. By allowing to write modular objects as twisted products of pairs of them, they open some new ways to express EPR and teleportation tasks. The appendix presents a look onto the rich structure of antilinear operator spaces. 1 2

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Section: Decompositions Of Normal Antilinear Operatorsmentioning

confidence: 99%

“…Hence their direct sum is a 2-graded algebra. An active domain of research are operator functions, for instance power series of elements of this algebra, see M. Huhtanen and A. Peramaki in [43,44] for example. Please notice: This rich theory is not under consideration here.…”

Section: Introductionmentioning

confidence: 99%

Anti- (conjugate) linearity

Uhlmann

2016

Sci. China Phys. Mech. Astron.

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An Inverse Spectral Problem for Non-Self-Adjoint Jacobi Matrices

Pushnitski,

Štampach

2024

International Mathematics Research Notices

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We consider the class of bounded symmetric Jacobi matrices $J$ with positive off-diagonal elements and complex diagonal elements. With each matrix $J$ from this class, we associate the spectral data, which consists of a pair $(\nu ,\psi )$. Here $\nu $ is the spectral measure of $|J|=\sqrt {J^{*}J}$ and $\psi $ is a phase function on the real line satisfying $|\psi |\leq 1$ almost everywhere with respect to the measure $\nu $. Our main result is that the map from $J$ to the pair $(\nu ,\psi )$ is a bijection between our class of Jacobi matrices and the set of all spectral data.

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Function theory and spectral mapping theorems for antilinear operators

Cited by 2 publications

References 17 publications

Anti- (conjugate) linearity

Anti- (conjugate) linearity

An Inverse Spectral Problem for Non-Self-Adjoint Jacobi Matrices

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