Normal matrices over hermitian discrete valuation rings

Adkins, William A.

doi:10.1016/0024-3795(91)90111-9

Cited by 7 publications

(10 citation statements)

References 4 publications

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“…The following lemma bears a strong resemblance to classical results [1,64], but since the precise formulation is slightly different in our paper, we provide a proof. Lemma 3.14.…”

Section: 3mentioning

confidence: 64%

“…Replace now e k and r e k`1 by e 1 k " 1 2 pe k `r e k`1 q and e 1 k`1 " 1 2 pe k ´r e k`1 q. This is an orthogonal base change that, by the claim, makes the order of xe 1 k , e 1 k y and…”

Section: 3mentioning

confidence: 99%

“…. , n 1 ´1u we have pM, λq " epn, k, , ξ, Fq, where n " n 1 ´ and k " minpk 1 ` , n 1 ´ q. (b) If pM 1 , λ 1 q " fpn 1 , k 1 , ξ, Fq, then pM, λq " f w ppn 1 ´a, n 1 ´bq, n 1 ´k1 ´a ´b, ξ, Fq.…”

Section: 4mentioning

confidence: 99%

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Twisted Blanchfield pairings and twisted signatures II: Relation to Casson-Gordon invariants

Borodzik¹,

Conway²,

Politarczyk³

2018

Preprint

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This paper decomposes into two main parts. In the algebraic part, we prove an isometry classification of linking forms over Rrt ˘1s and Crt ˘1s. Using this result, we associate signature functions to any such linking form and thoroughly investigate their properties. The topological part of the paper applies this machinery to twisted Blanchfield pairings of knots. We obtain twisted generalizations of the Levine-Tristram signature function which share several of its properties. We study the behavior of these twisted signatures under satellite operations. In the case of metabelian representations, we relate our invariants to the Casson-Gordon invariants and obtain a concrete formula for the metabelian Blanchfield pairings of satellites. Finally, we perform explicit computations on certain linear combinations of algebraic knots recovering a non-slice result of Hedden, Kirk and Livingston.

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“…The following lemma bears a strong resemblance to classical results [1,64], but since the precise formulation is slightly different in our paper, we provide a proof. Lemma 3.14.…”

Section: 3mentioning

confidence: 64%

Section: 3mentioning

confidence: 99%

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Twisted Blanchfield pairings and twisted signatures II: Relation to Casson-Gordon invariants

Borodzik¹,

Conway²,

Politarczyk³

2018

Preprint

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“…Its derivative exists a.e. and is bounded by the Lipschitz constant of A • c| [0,1] (with respect to the operator norm), by Claim 6.6. The assertion follows.…”

Section: 12mentioning

confidence: 91%

“…1.10] which exploits the monodromy of algebraic functions; see also [5, 3.5.1]. An algebraic version for normal matrices over so-called Hermitian discrete valuation rings is due to [1]. Actually, for C Q -curves of normal matrices, the local statement follows from [1], since the germs at 0 ∈ R of complex-valued C Q -functions form a Hermitian discrete valuation ring (as can be checked using Remark 5.1).…”

mentioning

confidence: 99%

Perturbation theory for normal operators

Rainer¹

2013

Trans. Amer. Math. Soc.

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Let $E \ni x\mapsto A(x)$ be a $\mathscr{C}$-mapping with values unbounded normal operators with common domain of definition and compact resolvent. Here $\mathscr{C}$ stands for $C^\infty$, $C^\omega$ (real analytic), $C^{[M]}$ (Denjoy--Carleman of Beurling or Roumieu type), $C^{0,1}$ (locally Lipschitz), or $C^{k,\alpha}$. The parameter domain $E$ is either $\mathbb R$ or $\mathbb R^n$ or an infinite dimensional convenient vector space. We completely describe the $\mathscr{C}$-dependence on $x$ of the eigenvalues and the eigenvectors of $A(x)$. Thereby we extend previously known results for self-adjoint operators to normal operators, partly improve them, and show that they are best possible. For normal matrices $A(x)$ we obtain partly stronger results.Comment: 32 pages, Remark 7.5 on m-sectorial operators added, accepted for publication in Trans. Amer. Math. So

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