Bounds for the counting function of the Jordan-Pólya numbers

Jean-Marie De Koninck; Nicolas Doyon; A. Arthur Bonkli Razafindrasoanaivolala; William Verreault

Displaying similar documents to “Bounds for the counting function of the Jordan-Pólya numbers”

On the Jordan model $C_{0}$ operators

Hari Bercovici (1977)

Studia Mathematica

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A Kleinecke-Shirokov type condition with Jordan automorphisms

Matej Brešar, Ajda Fošner, Maja Fošner (2001)

Studia Mathematica

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Let φ be a Jordan automorphism of an algebra . The situation when an element a ∈ satisfies $1 / 2 (φ (a) + φ^{- 1} (a)) = a$ is considered. The result which we obtain implies the Kleinecke-Shirokov theorem and Jacobson’s lemma.

On linked Jordan curves in $R^{3}$

M. Mateljević (1975)

Matematički Vesnik

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Lie algebras generated by Jordan operators

Peng Cao, Shanli Sun (2008)

Studia Mathematica

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It is proved that if $J_{i}$ is a Jordan operator on a Hilbert space with the Jordan decomposition $J_{i} = N_{i} + Q_{i}$ , where $N_{i}$ is normal and $Q_{i}$ is compact and quasinilpotent, i = 1,2, and the Lie algebra generated by J₁,J₂ is an Engel Lie algebra, then the Banach algebra generated by J₁,J₂ is an Engel algebra. Some results for normal operators and Jordan operators on Banach spaces are given.

Jordan isomorphisms and maps preserving spectra of certain operator products

Jinchuan Hou, Chi-Kwong Li, Ngai-Ching Wong (2008)

Studia Mathematica

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Let ₁, ₂ be (not necessarily unital or closed) standard operator algebras on locally convex spaces X₁, X₂, respectively. For k ≥ 2, consider different products $T ₁ * \dots * T_{k}$ on elements in $_{i}$ , which covers the usual product $T ₁ * \dots * T_{k} = T ₁ \dots T_{k}$ and the Jordan triple product T₁ ∗ T₂ = T₂T₁T₂. Let Φ: ₁ → ₂ be a (not necessarily linear) map satisfying $σ (Φ (A ₁) * \dots * Φ (A_{k})) = σ (A ₁ * \dots * A_{k})$ whenever any one of $A_{i}$ ’s has rank at most one. It is shown that if the range of Φ contains all rank one and rank two operators then Φ must be a Jordan isomorphism multiplied...

The operation $A B A$ in operator algebras

Marcell Gaál (2020)

Commentationes Mathematicae Universitatis Carolinae

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The binary operation $a b a$ , called Jordan triple product, and its variants (such as e.g. the sequential product $\sqrt{a} b \sqrt{a}$ or the inverted Jordan triple product $a b^{- 1} a$ ) appear in several branches of operator theory and matrix analysis. In this paper we briefly survey some analytic and algebraic properties of these operations, and investigate their intimate connection to Thompson type isometries in different operator algebras.

Equidecomposability of Jordan domains under groups of isometries

M. Laczkovich (2003)

Fundamenta Mathematicae

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Let $G_{d}$ denote the isometry group of $ℝ^{d}$ . We prove that if G is a paradoxical subgroup of $G_{d}$ then there exist G-equidecomposable Jordan domains with piecewise smooth boundaries and having different volumes. On the other hand, we construct a system $ℱ_{d}$ of Jordan domains with differentiable boundaries and of the same volume such that $ℱ_{d}$ has the cardinality of the continuum, and for every amenable subgroup G of $G_{d}$ , the elements of $ℱ_{d}$ are not G-equidecomposable; moreover, their interiors are not G-equidecomposable...

Product of operators and numerical range preserving maps

Chi-Kwong Li, Nung-Sing Sze (2006)

Studia Mathematica

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Let V be the C*-algebra B(H) of bounded linear operators acting on the Hilbert space H, or the Jordan algebra S(H) of self-adjoint operators in B(H). For a fixed sequence (i₁, ..., iₘ) with i₁, ..., iₘ ∈ 1, ..., k, define a product of $A ₁, . . ., A_{k} \in V$ by $A ₁ * \dots * A_{k} = A_{i ₁} \dots A_{i ₘ}$ . This includes the usual product $A ₁ * \dots * A_{k} = A ₁ \dots A_{k}$ and the Jordan triple product A*B = ABA as special cases. Denote the numerical range of A ∈ V by W(A) = (Ax,x): x ∈ H, (x,x) = 1. If there is a unitary operator U and a scalar μ satisfying $μ^{m} = 1$ such that ϕ: V → V has...

On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces

Mikio Kato, Lech Maligranda, Yasuji Takahashi (2001)

Studia Mathematica

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Some relations between the James (or non-square) constant J(X) and the Jordan-von Neumann constant $C_{N J} (X)$ , and the normal structure coefficient N(X) of Banach spaces X are investigated. Relations between J(X) and J(X*) are given as an answer to a problem of Gao and Lau [16]. Connections between $C_{N J} (X)$ and J(X) are also shown. The normal structure coefficient of a Banach space is estimated by the $C_{N J} (X)$ -constant, which implies that a Banach space with $C_{N J} (X)$ -constant less than 5/4 has the fixed point property. ...

A Menon-type identity using Klee's function

Arya Chandran, Neha Elizabeth Thomas, K. Vishnu Namboothiri (2022)

Czechoslovak Mathematical Journal

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Menon’s identity is a classical identity involving gcd sums and the Euler totient function $φ$ . A natural generalization of $φ$ is the Klee’s function $Φ_{s}$ . We derive a Menon-type identity using Klee’s function and a generalization of the gcd function. This identity generalizes an identity given by Y. Li and D. Kim (2017).

Non-hyperreflexive reflexive spaces of operators

Roman V. Bessonov, Janko Bračič, Michal Zajac (2011)

Studia Mathematica

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We study operators whose commutant is reflexive but not hyperreflexive. We construct a C₀ contraction and a Jordan block operator $S_{B}$ associated with a Blaschke product B which have the above mentioned property. A sufficient condition for hyperreflexivity of $S_{B}$ is given. Some other results related to hyperreflexivity of spaces of operators that could be interesting in themselves are proved.

Generalized Higher Derivations on Lie Ideals of Triangular Algebras

Mohammad Ashraf, Nazia Parveen, Bilal Ahmad Wani (2017)

Communications in Mathematics

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Let $𝔄 = (\begin{matrix} 𝒜 & ℳ \\ ℬ \end{matrix})$ be the triangular algebra consisting of unital algebras $𝒜$ and $ℬ$ over a commutative ring $R$ with identity $1$ and $ℳ$ be a unital $(𝒜, ℬ)$ -bimodule. An additive subgroup $𝔏$ of $𝔄$ is said to be a Lie ideal of $𝔄$ if $[𝔏, 𝔄] \subseteq 𝔏$ . A non-central square closed Lie ideal $𝔏$ of $𝔄$ is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on $𝔄$ , every generalized Jordan triple higher derivation of $𝔏$ into $𝔄$ is a generalized higher derivation of $𝔏$ into $𝔄$ . ...

Product decompositions of quasirandom groups and a Jordan type theorem

Nikolay Nikolov, László Pyber (2011)

Journal of the European Mathematical Society

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We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$ , then for every subset $B$ of $G$ with $| B | > | G | / k^{1 / 3}$ we have $B^{3} = G$ . We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if $k \geq 2$ , then $G$ has a...

On the characterization of certain additive maps in prime $*$ -rings

Mohammad Ashraf, Mohammad Aslam Siddeeque, Abbas Hussain Shikeh (2024)

Czechoslovak Mathematical Journal

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Let $𝒜$ be a noncommutative prime ring equipped with an involution ‘ $*$ ’, and let $𝒬_{m s} (𝒜)$ be the maximal symmetric ring of quotients of $𝒜$ . Consider the additive maps $ℋ$ and $𝒯 : 𝒜 \to 𝒬_{m s} (𝒜)$ . We prove the following under some inevitable torsion restrictions. (a) If $m$ and $n$ are fixed positive integers such that $(m + n) 𝒯 (a^{2}) = m 𝒯 (a) a^{*} + n a 𝒯 (a)$ for all $a \in 𝒜$ and $(m + n) ℋ (a^{2}) = m ℋ (a) a^{*} + n a 𝒯 (a)$ for all $a \in 𝒜$ , then $ℋ = 0$ . (b) If $𝒯 (a b a) = a 𝒯 (b) a^{*}$ for all $a, b \in 𝒜$ , then $𝒯 = 0$ . Furthermore, we characterize Jordan left $τ$ -centralizers in semiprime rings admitting an anti-automorphism $τ$ . As applications, we find the...

Jordan types for indecomposable modules of finite group schemes

Rolf Farnsteiner (2014)

Journal of the European Mathematical Society

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In this article we study the interplay between algebro-geometric notions related to $π$ -points and structural features of the stable Auslander-Reiten quiver of a finite group scheme. We show that $π$ -points give rise to a number of new invariants of the AR-quiver on one hand, and exploit combinatorial properties of AR-components to obtain information on $π$ -points on the other. Special attention is given to components containing Carlson modules, constantly supported modules, and endo-trivial...

Quadro-quadric Cremona transformations in low dimensions via the $J C$ -correspondence

Luc Pirio, Francesco Russo (2014)

Annales de l’institut Fourier

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It has been previously established that a Cremona transformation of bidegree (2,2) is linearly equivalent to the projectivization of the inverse map of a rank 3 Jordan algebra. We call this result the “”. In this article, we apply it to the study of quadro-quadric Cremona transformations in low-dimensional projective spaces. In particular we describe new very simple families of such birational maps and obtain complete and explicit classifications in dimension 4 and 5.

On local CR-transformations of Levi-degenerate group orbits in compact Hermitian symmetric spaces

Wilhelm Kaup, Dmitri Zaitsev (2006)

Journal of the European Mathematical Society

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We present a large class of homogeneous 2-nondegenerate CR-manifolds $M$ , both of hypersurface type and of arbitrarily high CR-codimension, with the following property: Every CR-equivalence between domains $U$ , $V$ in $M$ extends to a global real-analytic CR-automorphism of $M$ . We show that this class contains $G$ -orbits in Hermitian symmetric spaces $Z$ of compact type, where $G$ is a real form of the complex Lie group $Aut {(Z)}^{0}$ and $G$ has an open orbit that is a bounded symmetric domain of tube type. ...

Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions

G. Letac, J. Wesołowski (2011)

Bulletin de la Société Mathématique de France

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If the space $𝒬$ of quadratic forms in $ℝ^{n}$ is splitted in a direct sum $𝒬_{1} \oplus ... \oplus 𝒬_{k}$ and if $X$ and $Y$ are independent random variables of $ℝ^{n}$ , assume that there exist a real number $a$ such that $E (X | X + Y) = a (X + Y)$ and real distinct numbers $b_{1}, . . ., b_{k}$ such that $E (q (X) | X + Y) = b_{i} q (X + Y)$ for any $q$ in $𝒬_{i} .$ We prove that this happens only when $k = 2$ , when $ℝ^{n}$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure.

On a theorem of Lindelof

Vladimir Ya. Gutlyanskii, Olli Martio, Vladimir Ryazanov (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We give a quasiconformal version of the proof for the classical Lindelof theorem: Let $f$ map the unit disk $𝔻$ conformally onto the inner domain of a Jordan curve $𝒞$ : Then $𝒞$ is smooth if and only if arg $f^{'} (z)$ has a continuous extension to $\bar{𝔻}$ . Our proof does not use the Poisson integral representation of harmonic functions in the unit disk.

On a modification of the Poisson integral operator

Dariusz Partyka (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Given a quasisymmetric automorphism $γ$ of the unit circle $𝕋$ we define and study a modification $P_{γ}$ of the classical Poisson integral operator in the case of the unit disk $𝔻$ . The modification is done by means of the generalized Fourier coefficients of $γ$ . For a Lebesgue’s integrable complexvalued function $f$ on $𝕋$ , $P_{γ} [f]$ is a complex-valued harmonic function in $𝔻$ and it coincides with the classical Poisson integral of $f$ provided $γ$ is the identity mapping on $𝕋$ . Our considerations are motivated by...

$(φ, ϕ)$ -derivations on semiprime rings and Banach algebras

Bilal Ahmad Wani (2021)

Communications in Mathematics

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Let $ℛ$ be a semiprime ring with unity $e$ and $φ$ , $ϕ$ be automorphisms of $ℛ$ . In this paper it is shown that if $ℛ$ satisfies $2 𝒟 (x^{n}) = 𝒟 (x^{n - 1}) φ (x) + ϕ (x^{n - 1}) 𝒟 (x) + 𝒟 (x) φ (x^{n - 1}) + ϕ (x) 𝒟 (x^{n - 1})$ for all $x \in ℛ$ and some fixed integer $n \geq 2$ , then $𝒟$ is an ( $φ$ , $ϕ$ )-derivation. Moreover, this result makes it possible to prove that if $ℛ$ admits an additive mappings $𝒟, 𝒢 : ℛ \to ℛ$ satisfying the relations $\begin{matrix} 2 𝒟 (x^{n}) = 𝒟 (x^{n - 1}) φ (x) + ϕ (x^{n - 1}) 𝒢 (x) + 𝒢 (x) φ (x^{n - 1}) + ϕ (x) 𝒢 (x^{n - 1}), \\ 2 𝒢 (x^{n}) = 𝒢 (x^{n - 1}) φ (x) + ϕ (x^{n - 1}) 𝒟 (x) + 𝒟 (x) φ (x^{n - 1}) + ϕ (x) 𝒟 (x^{n - 1}), \end{matrix}$ for all $x \in ℛ$ and some fixed integer $n \geq 2$ , then $𝒟$ and $𝒢$ are ( $φ$ , $ϕ$ )derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras. ...

On finite groups acting on a connected sum of 3-manifolds S² × S¹

Bruno P. Zimmermann (2014)

Fundamenta Mathematicae

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Let $H_{g}$ denote the closed 3-manifold obtained as the connected sum of g copies of S² × S¹, with free fundamental group of rank g. We prove that, for a finite group G acting on $H_{g}$ which induces a faithful action on the fundamental group, there is an upper bound for the order of G which is quadratic in g, but there does not exist a linear bound in g. This implies then a Jordan-type bound for arbitrary finite group actions on $H_{g}$ which is quadratic in g. For the proofs we develop a calculus...

Lower semicontinuous envelopes in $W^{1, 1} \times L^{p}$

Ana Margarida Ribeiro, Elvira Zappale (2014)

Banach Center Publications

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The lower semicontinuity of functionals of the type $\int_{Ω} f (x, u, v, \nabla u) d x$ with respect to the $(W^{1, 1} \times L^{p})$ -weak* topology is studied. Moreover, in absence of lower semicontinuity, an integral representation in $W^{1, 1} \times L^{p}$ for the lower semicontinuous envelope is also provided.