On a Weyl–von Neumann type theorem for antilinear self-adjoint operators

Ruotsalainen, Santtu

doi:10.4064/sm213-3-1

Cited by 5 publications

(3 citation statements)

References 18 publications

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“…|f | is a positive linear operator and in case that f is invertible, U is antiunitary. A good reference on the subject of antilinear and antiunitary operators is the paper by Routsalainen [20].…”

Section: The Special Jordan Algebra Of Hilbert Space Operatorsmentioning

confidence: 99%

On the structure group of an infinite dimensional JB-algebra

Larotonda¹,

Luna²

2022

Preprint

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We extend several results for the structure group of a real Jordan algebra V, to the setting of infinite dimensional JB-algebras. We prove that the structure group Str(V), the cone preserving group G(Ω) and the automorphism group Aut(V) of the algebra V are embedded Banach-Lie groups of GL(V), and that each of the inclusions Aut(V) ⊂ G(Ω) ⊂ Str(V) are of embedded Banach-Lie subgroups. We give a full description of the components of Str(V) via cones, isotopes and central projections. We apply these results to V = B(H) sa the special JB-algebra of self-adjoint operators on an infinite dimensional complex Hilbert space, describing the groups Str(V), G(Ω), Aut(V), their Banach-Lie algebras and their connected components. We show that the action of the unitary group of H on Aut(V) has smooth local cross sections, thus Aut(V) is a smooth principal bundle over the unitary group, with structure group S 1 .

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Section: The Special Jordan Algebra Of Hilbert Space Operatorsmentioning

confidence: 99%

On the structure group of an infinite dimensional JB-algebra

Larotonda¹,

Luna²

2022

Preprint

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“…Thus, there is a straightforward dictionary between conjugate-linear operators that are R-selfadjoint and C-symmetric operators. A study of the first class, motivated by classical examples such as Beltrami or Hankel operators, has been vigorously pursued by the Finnish school [39,[80][81][82][83]128]. We confine ourselves to reproduce below only a small portion of their results.…”

Section: Lemma 73 Consider the Following Block-matrix Operator H And ...mentioning

confidence: 99%

Mathematical and physical aspects of complex symmetric operators

Garcia¹,

Prodan²,

Putinar³

2014

J. Phys. A: Math. Theor.

138

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ABSTRACT. Recent advances in the theory of complex symmetric operators are presented and related to current studies in non-hermitian quantum mechanics. The main themes of the survey are: the structure of complex symmetric operators, C-selfadjoint extensions of C-symmetric unbounded operators, resolvent estimates, reality of spectrum, bases of C-orthonormal vectors, and conjugatelinear symmetric operators. The main results are complemented by a variety of natural examples arising in field theory, quantum physics, and complex variables.

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“…By Corollary 3.7, for each n there exists a biradial measure µ n such that the corresponding Jacobi parameters are {α j } n j=1 , {β j } n−1 j=1 , and we denote the corresponding complex Jacobi matrix by J We next obtain a spectral theorem for bounded antilinear self-adjoint operators. For an approach using spectral integrals, see [20].…”

Section: Biradial Measures With Infinite Support and Antilinear Jacobmentioning

confidence: 99%

Function theory and spectral mapping theorems for antilinear operators

Huhtanen¹,

Perämäki²

2014

J. Operator Theory

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Unlike in complex linear operator theory, polynomials or, more generally, Laurent series in antilinear operators cannot be modelled with complex analysis. There exists a corresponding function space, though, surfacing in spectral mapping theorems. These spectral mapping theorems are inclusive in general. Equality can be established in the self-adjoint case. The arising functions are shown to possess a biradial character. It is shown that to any given set of Jacobi parameters corresponds a biradial measure yielding these parameters in an iterative orthogonalization process in this function space, once equipped with the corresponding L 2 structure.

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On a Weyl–von Neumann type theorem for antilinear self-adjoint operators

Cited by 5 publications

References 18 publications

On the structure group of an infinite dimensional JB-algebra

On the structure group of an infinite dimensional JB-algebra

Mathematical and physical aspects of complex symmetric operators

Function theory and spectral mapping theorems for antilinear operators

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