Rota-type operators on 3-dimensional nilpotent associative algebras

N.G. Abdujabborov; I.A. Karimjanov and M.A. Kodirova

Displaying similar documents to “Rota-type operators on 3-dimensional nilpotent associative algebras”

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. ...

On nilpotent operators

Laura Burlando (2005)

Studia Mathematica

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We give several necessary and sufficient conditions in order that a bounded linear operator on a Banach space be nilpotent. We also discuss three necessary conditions for nilpotency. Furthermore, we construct an infinite family (in one-to-one correspondence with the square-summable sequences ${(ε ₙ)}_{n \in ℕ}$ of strictly positive real numbers) of nonnilpotent quasinilpotent operators on an infinite-dimensional Hilbert space, all the iterates of each of which have closed range. Each of these operators...

Leibniz $A$ -algebras

David A. Towers (2020)

Communications in Mathematics

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A finite-dimensional Lie algebra is called an $A$ -algebra if all of its nilpotent subalgebras are abelian. These arise in the study of constant Yang-Mills potentials and have also been particularly important in relation to the problem of describing residually finite varieties. They have been studied by several authors, including Bakhturin, Dallmer, Drensky, Sheina, Premet, Semenov, Towers and Varea. In this paper we establish generalisations of many of these results to Leibniz algebras. ...

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.

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.

Standardly stratified split and lower triangular algebras

Eduardo do N. Marcos, Hector A. Merklen, Corina Sáenz (2002)

Colloquium Mathematicae

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In the first part, we study algebras A such that A = R ⨿ I, where R is a subalgebra and I a two-sided nilpotent ideal. Under certain conditions on I, we show that A is standardly stratified if and only if R is standardly stratified. Next, for $A = [\begin{matrix} U & 0 \\ M & V \end{matrix}]$ , we show that A is standardly stratified if and only if the algebra R = U × V is standardly stratified and $_{V} M$ is a good V-module.

Engel BCI-algebras: an application of left and right commutators

Ardavan Najafi, Arsham Borumand Saeid (2021)

Mathematica Bohemica

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We introduce Engel elements in a BCI-algebra by using left and right normed commutators, and some properties of these elements are studied. The notion of $n$ -Engel BCI-algebra as a natural generalization of commutative BCI-algebras is introduced, and we discuss Engel BCI-algebra, which is defined by left and right normed commutators. In particular, we prove that any nilpotent BCI-algebra of type $2$ is an Engel BCI-algebra, but solvable BCI-algebras are not Engel, generally. Also, it is...

On operators with the same local spectra

Aleksandar Torgašev (1998)

Czechoslovak Mathematical Journal

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Let $B (X)$ be the algebra of all bounded linear operators in a complex Banach space $X$ . We consider operators $T_{1}, T_{2} \in B (X)$ satisfying the relation $σ_{T_{1}} (x) = σ_{T_{2}} (x)$ for any vector $x \in X$ , where $σ_{T} (x)$ denotes the local spectrum of $T \in B (X)$ at the point $x \in X$ . We say then that $T_{1}$ and $T_{2}$ have the same local spectra. We prove that then, under some conditions, $T_{1} - T_{2}$ is a quasinilpotent operator, that is $∥ {(T_{1} - T_{2})}^{n} ∥^{1 / n} \to 0$ as $n \to \infty$ . Without these conditions, we describe the operators with the same local spectra only in some particular cases.

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.

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 ∈...

Альтернативные алгебры с тождеством ${[x, y]}^{n} = 0$

И.П. Шестаков (1981)

Algebra i Logika

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Besov spaces and 2-summing operators

M. A. Fugarolas (2004)

Colloquium Mathematicae

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Let Π₂ be the operator ideal of all absolutely 2-summing operators and let $I_{m}$ be the identity map of the m-dimensional linear space. We first establish upper estimates for some mixing norms of $I_{m}$ . Employing these estimates, we study the embedding operators between Besov function spaces as mixing operators. The result obtained is applied to give sufficient conditions under which certain kinds of integral operators, acting on a Besov function space, belong to Π₂; in this context, we also...

On some properties of the upper central series in Leibniz algebras

Leonid A. Kurdachenko, Javier Otal, Igor Ya. Subbotin (2019)

Commentationes Mathematicae Universitatis Carolinae

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This article discusses the Leibniz algebras whose upper hypercenter has finite codimension. It is proved that such an algebra $L$ includes a finite dimensional ideal $K$ such that the factor-algebra $L / K$ is hypercentral. This result is an extension to the Leibniz algebra of the corresponding result obtained earlier for Lie algebras. It is also analogous to the corresponding results obtained for groups and modules.

Multiple summing operators on $l_{p}$ spaces

Dumitru Popa (2014)

Studia Mathematica

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We use the Maurey-Rosenthal factorization theorem to obtain a new characterization of multiple 2-summing operators on a product of $l_{p}$ spaces. This characterization is used to show that multiple s-summing operators on a product of $l_{p}$ spaces with values in a Hilbert space are characterized by the boundedness of a natural multilinear functional (1 ≤ s ≤ 2). We use these results to show that there exist many natural multiple s-summing operators $T : l_{4 / 3} \times l_{4 / 3} \to l ₂$ such that none of the associated linear operators...

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.