On nilpotent operators

Laura Burlando

Displaying similar documents to “On nilpotent operators”

Parallelepipeds, nilpotent groups and Gowers norms

Bernard Host, Bryna Kra (2008)

Bulletin de la Société Mathématique de France

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In his proof of Szemerédi’s Theorem, Gowers introduced certain norms that are defined on a parallelepiped structure. A natural question is on which sets a parallelepiped structure (and thus a Gowers norm) can be defined. We focus on dimensions $2$ and $3$ and show when this possible, and describe a correspondence between the parallelepiped structures and nilpotent groups.

Convergence of formal solutions of first order singular partial differential equations of nilpotent type

Masatake Miyake, Akira Shirai (2012)

Banach Center Publications

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Let (x,y,z) ∈ ℂ³. In this paper we shall study the solvability of singular first order partial differential equations of nilpotent type by the following typical example: $P u (x, y, z) : = (y \partial_{x} - z \partial_{y}) u (x, y, z) = f (x, y, z) \in_{x, y, z}$ , where $P = y \partial_{x} - z \partial_{y} :_{x, y, z} \to_{x, y, z}$ . For this equation, our aim is to characterize the solvability on $_{x, y, z}$ by using the Im P, Coker P and Ker P, and we give the exact forms of these sets.

Partial differential equations in Banach spaces involving nilpotent linear operators

Antonia Chinnì, Paolo Cubiotti (1996)

Annales Polonici Mathematici

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Let E be a Banach space. We consider a Cauchy problem of the type ⎧ $D_{t}^{k} u + \sum_{j = 0}^{k - 1} \sum_{| α | \leq m} A_{j, α} (D_{t}^{j} D_{x}^{α} u) = f$ in $ℝ^{n + 1}$ , ⎨ ⎩ $D_{t}^{j} u (0, x) = φ_{j} (x)$ in $ℝ^{n}$ , j=0,...,k-1, where each $A_{j, α}$ is a given continuous linear operator from E into itself. We prove that if the operators $A_{j, α}$ are nilpotent and pairwise commuting, then the problem is well-posed in the space of all functions $u \in C^{\infty} (ℝ^{n + 1}, E)$ whose derivatives are equi-bounded on each bounded subset of $ℝ^{n + 1}$ .

Quasinilpotent operators in operator Lie algebras II

Peng Cao (2009)

Studia Mathematica

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In this paper, it is proved that the Banach algebra $\bar{(ℒ)}$ , generated by a Lie algebra ℒ of operators, consists of quasinilpotent operators if ℒ consists of quasinilpotent operators and $\bar{(ℒ)}$ consists of polynomially compact operators. It is also proved that $\bar{(ℒ)}$ consists of quasinilpotent operators if ℒ is an essentially nilpotent Engel Lie algebra generated by quasinilpotent operators. Finally, Banach algebras generated by essentially nilpotent Lie algebras are shown to be compactly quasinilpotent. ...

Hurewicz-Serre theorem in extension theory

M. Cencelj, J. Dydak, A. Mitra, A. Vavpetič (2008)

Fundamenta Mathematicae

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The paper is devoted to generalizations of the Cencelj-Dranishnikov theorems relating extension properties of nilpotent CW complexes to their homology groups. Here are the main results of the paper: Theorem 0.1. Let L be a nilpotent CW complex and F the homotopy fiber of the inclusion i of L into its infinite symmetric product SP(L). If X is a metrizable space such that $X τ K (H_{k} (L), k)$ for all k ≥ 1, then $X τ K (π_{k} (F), k)$ and $X τ K (π_{k} (L), k)$ for all k ≥ $.$ Theorem 0.2. Let X be a metrizable space such that dim(X) < ∞ or X ∈...

Images of locally nilpotent derivations of bivariate polynomial algebras over a domain

Xiaosong Sun, Beini Wang (2024)

Czechoslovak Mathematical Journal

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We study the LND conjecture concerning the images of locally nilpotent derivations, which arose from the Jacobian conjecture. Let $R$ be a domain containing a field of characteristic zero. We prove that, when $R$ is a one-dimensional unique factorization domain, the image of any locally nilpotent $R$ -derivation of the bivariate polynomial algebra $R [x, y]$ is a Mathieu-Zhao subspace. Moreover, we prove that, when $R$ is a Dedekind domain, the image of a locally nilpotent $R$ -derivation of $R [x, y]$ with some...

Commutants and derivation ranges

Salah Mecheri (1999)

Czechoslovak Mathematical Journal

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In this paper we obtain some results concerning the set $ℳ = \cup \{\bar{R (δ_{A})} \cap {A}^{'} A \in ℒ (H)\}$ , where $\bar{R (δ_{A})}$ is the closure in the norm topology of the range of the inner derivation $δ_{A}$ defined by $δ_{A} (X) = A X - X A .$ Here $ℋ$ stands for a Hilbert space and we prove that every compact operator in ${\bar{R (δ_{A})}}^{w} \cap {A^{*}}^{'}$ is quasinilpotent if $A$ is dominant, where ${\bar{R (δ_{A})}}^{w}$ is the closure of the range of $δ_{A}$ in the weak topology.

Classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center

Bin Ren, Lin Sheng Zhu (2017)

Czechoslovak Mathematical Journal

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A Lie algebra $L$ is called 2-step nilpotent if $L$ is not abelian and $[L, L]$ lies in the center of $L$ . 2-step nilpotent Lie algebras are useful in the study of some geometric problems, and their classification has been an important problem in Lie theory. In this paper, we give a classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center.

Finite groups in which every self-centralizing subgroup is nilpotent or subnormal or a TI-subgroup

Jiangtao Shi, Na Li (2021)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group. We prove that if every self-centralizing subgroup of $G$ is nilpotent or subnormal or a TI-subgroup, then every subgroup of $G$ is nilpotent or subnormal. Moreover, $G$ has either a normal Sylow $p$ -subgroup or a normal $p$ -complement for each prime divisor $p$ of $| G |$ .

$𝒟_{n, r}$ is not potentially nilpotent for $n \geq 4 r - 2$

Yan Ling Shao, Yubin Gao, Wei Gao (2016)

Czechoslovak Mathematical Journal

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An $n \times n$ sign pattern $𝒜$ is said to be potentially nilpotent if there exists a nilpotent real matrix $B$ with the same sign pattern as $𝒜$ . Let $𝒟_{n, r}$ be an $n \times n$ sign pattern with $2 \leq r \leq n$ such that the superdiagonal and the $(n, n)$ entries are positive, the $(i, 1)$ $(i = 1, \dots, r)$ and $(i, i - r + 1)$ $(i = r + 1, \dots, n)$ entries are negative, and zeros elsewhere. We prove that for $r \geq 3$ and $n \geq 4 r - 2$ , the sign pattern $𝒟_{n, r}$ is not potentially nilpotent, and so not spectrally arbitrary.

Spaces of operators and c₀

P. Lewis (2001)

Studia Mathematica

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Bessaga and Pełczyński showed that if c₀ embeds in the dual X* of a Banach space X, then ℓ¹ embeds complementably in X, and $ℓ^{\infty}$ embeds as a subspace of X*. In this note the Diestel-Faires theorem and techniques of Kalton are used to show that if X is an infinite-dimensional Banach space, Y is an arbitrary Banach space, and c₀ embeds in L(X,Y), then $ℓ^{\infty}$ embeds in L(X,Y), and ℓ¹ embeds complementably in $X \otimes_{γ} Y *$ . Applications to embeddings of c₀ in various spaces of operators are given.

Leibniz's rule on two-step nilpotent Lie groups

Krystian Bekała (2016)

Colloquium Mathematicae

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Let be a nilpotent Lie algebra which is also regarded as a homogeneous Lie group with the Campbell-Hausdorff multiplication. This allows us to define a generalized multiplication $f g = {(f^{\lor} * g^{\lor})}^{\land}$ of two functions in the Schwartz class (*), where $^{\lor}$ and $^{\land}$ are the Abelian Fourier transforms on the Lie algebra and on the dual * and ∗ is the convolution on the group . In the operator analysis on nilpotent Lie groups an important notion is the one of symbolic calculus which can be viewed as a higher order...

On the nilpotent residuals of all subalgebras of Lie algebras

Wei Meng, Hailou Yao (2018)

Czechoslovak Mathematical Journal

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Let $𝒩$ denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra $L$ over an arbitrary field $𝔽$ , there exists a smallest ideal $I$ of $L$ such that $L / I \in 𝒩$ . This uniquely determined ideal of $L$ is called the nilpotent residual of $L$ and is denoted by $L^{𝒩}$ . In this paper, we define the subalgebra $S (L) = ⋂_{H \leq L} I_{L} (H^{𝒩})$ . Set $S_{0} (L) = 0$ . Define $S_{i + 1} (L) / S_{i} (L) = S (L / S_{i} (L))$ for $i \geq 1$ . By $S_{\infty} (L)$ denote the terminal term of the ascending series. It is proved that $L = S_{\infty} (L)$ if and only if $L^{𝒩}$ is nilpotent. In addition, we investigate the basic properties of a...

The evolution and Poisson kernels on nilpotent meta-abelian groups

Richard Penney, Roman Urban (2013)

Studia Mathematica

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Let S be a semidirect product S = N⋊ A where N is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and A is isomorphic to $ℝ^{k}$ , k>1. We consider a class of second order left-invariant differential operators on S of the form $ℒ_{α} = L^{a} + Δ_{α}$ , where $α \in ℝ^{k}$ , and for each $a \in ℝ^{k}, L^{a}$ is left-invariant second order differential operator on N and $Δ_{α} = Δ - ⟨ α, \nabla ⟩$ , where Δ is the usual Laplacian on $ℝ^{k}$ . Using some probabilistic techniques (e.g., skew-product formulas for diffusions on S and N respectively)...