Linear combinations of generators in multiplicatively invariant spaces

Victoria Paternostro

Displaying similar documents to “Linear combinations of generators in multiplicatively invariant spaces”

On invariant, dual invariant and absolute formulas

Andrzej Mostowski

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CONTENTS Introduction..............................................................................................................................................................3 1. Lemmas concerning first order formulas.....................................................................................................5 2. Representability of recursively enumerable sets........................................................................................9 3. Simple theory of types.......................................................................................................................................10...

On a translation property of positive definite functions

Lars Omlor, Michael Leinert (2010)

Banach Center Publications

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If G is a locally compact group with a compact invariant neighbourhood of the identity e, the following property (*) holds: For every continuous positive definite function h≥ 0 with compact support there is a constant $C_{h} > 0$ such that $\int L_{x} h \cdot g \leq C_{h} \int h g$ for every continuous positive definite g≥0, where $L_{x}$ is left translation by x. In [L], property (*) was stated, but the above inequality was proved for special h only. That “for one h” implies “for all h” seemed obvious, but turned out not to be obvious at...

A finite multiplicity Helson-Lowdenslager-de Branges theorem

Sneh Lata, Meghna Mittal, Dinesh Singh (2010)

Studia Mathematica

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We prove two theorems. The first theorem reduces to a scalar situation the well known vector-valued generalization of the Helson-Lowdenslager theorem that characterizes the invariant subspaces of the operator of multiplication by the coordinate function z on the vector-valued Lebesgue space L²(;ℂⁿ). Our approach allows us to prove an equivalent version of the vector-valued Helson-Lowdenslager theorem in a completely scalar setting, thereby eliminating the use of range functions and partial...

Automatic continuity of operators commuting with translations

J. Alaminos, J. Extremera, A. R. Villena (2006)

Studia Mathematica

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Let $τ_{X}$ and $τ_{Y}$ be representations of a topological group G on Banach spaces X and Y, respectively. We investigate the continuity of the linear operators Φ: X → Y with the property that $Φ \circ τ_{X} (t) = τ_{Y} (t) \circ Φ$ for each t ∈ G in terms of the invariant vectors in Y and the automatic continuity of the invariant linear functionals on X.

The almost Daugavet property and translation-invariant subspaces

Simon Lücking (2014)

Colloquium Mathematicae

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Let G be a metrizable, compact abelian group and let Λ be a subset of its dual group Ĝ. We show that $C_{Λ} (G)$ has the almost Daugavet property if and only if Λ is an infinite set, and that $L ¹_{Λ} (G)$ has the almost Daugavet property if and only if Λ is not a Λ(1) set.

On the Finsler geometry of the Heisenberg group $H_{2 n + 1}$ and its extension

Mehri Nasehi (2021)

Archivum Mathematicum

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We first classify left invariant Douglas $(α, β)$ -metrics on the Heisenberg group $H_{2 n + 1}$ of dimension $2 n + 1$ and its extension i.e., oscillator group. Then we explicitly give the flag curvature formulas and geodesic vectors for these spaces, when equipped with these metrics. We also explicitly obtain $S$ -curvature formulas of left invariant Randers metrics of Douglas type on these spaces and obtain a comparison on geometry of these spaces, when equipped with left invariant Douglas $(α, β)$ -metrics. More exactly,...

Invariants of finite groups generated by generalized transvections in the modular case

Xiang Han, Jizhu Nan, Chander K. Gupta (2017)

Czechoslovak Mathematical Journal

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We investigate the invariant rings of two classes of finite groups $G \leq GL (n, F_{q})$ which are generated by a number of generalized transvections with an invariant subspace $H$ over a finite field $F_{q}$ in the modular case. We name these groups generalized transvection groups. One class is concerned with a given invariant subspace which involves roots of unity. Constructing quotient groups and tensors, we deduce the invariant rings and study their Cohen-Macaulay and Gorenstein properties. The other is concerned...

Geodesically equivalent metrics on homogenous spaces

Neda Bokan, Tijana Šukilović, Srdjan Vukmirović (2019)

Czechoslovak Mathematical Journal

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Two metrics on a manifold are geodesically equivalent if the sets of their unparameterized geodesics coincide. We show that if two $G$ -invariant metrics of arbitrary signature on homogenous space $G / H$ are geodesically equivalent, they are affinely equivalent, i.e. they have the same Levi-Civita connection. We also prove that the existence of nonproportional, geodesically equivalent, $G$ -invariant metrics on homogenous space $G / H$ implies that their holonomy algebra cannot be full. We give an algorithm...

Shift invariant operators and a saturation theorem

Karol Dziedziul (2003)

Applicationes Mathematicae

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The properties of shift invariant operators $Q_{h}$ are proved: It is shown that Q has polynomial order r iff r is the rate of convergence of $Q_{h}$ . A weak saturation theorem is given. If f is replaced by $Q_{f} h$ in the weak saturation formula the asymptotics of the expression is calculated. Moreover, bootstrap approximation is introduced.

A note on integer translates of a square integrable function on ℝ

Maciej Paluszyński (2010)

Colloquium Mathematicae

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We consider the subspace of L²(ℝ) spanned by the integer shifts of one function ψ, and formulate a condition on the family ${ψ (\cdot - n)}_{n = - \infty}^{\infty}$ , which is equivalent to the weight function $\sum_{n = - \infty}^{\infty} | ψ ̂ (\cdot + n) | ²$ being > 0 a.e.

Stable invariant subspaces for operators on Hilbert space

John B. Conway, Don Hadwin (1997)

Annales Polonici Mathematici

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If T is a bounded operator on a separable complex Hilbert space ℋ, an invariant subspace ℳ for T is stable provided that whenever $T_{n}$ is a sequence of operators such that $‖ T_{n} - T ‖ \to 0$ , there is a sequence of subspaces $ℳ_{n}$ , with $ℳ_{n}$ in $L a t T_{n}$ for all n, such that $P_{ℳ_{n}} \to P_{ℳ}$ in the strong operator topology. If the projections converge in norm, ℳ is called a norm stable invariant subspace. This paper characterizes the stable invariant subspaces of the unilateral shift of finite multiplicity and normal operators. It also...

Vector invariant ideals of abelian group algebras under the actions of the unitary groups and orthogonal groups

Lingli Zeng, Jizhu Nan (2016)

Czechoslovak Mathematical Journal

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Let $F$ be a finite field of characteristic $p$ and $K$ a field which contains a primitive $p$ th root of unity and $char K \neq p$ . Suppose that a classical group $G$ acts on the $F$ -vector space $V$ . Then it can induce the actions on the vector space $V \oplus V$ and on the group algebra $K [V \oplus V]$ , respectively. In this paper we determine the structure of $G$ -invariant ideals of the group algebra $K [V \oplus V]$ , and establish the relationship between the invariant ideals of $K [V]$ and the vector invariant ideals of $K [V \oplus V]$ , if $G$ is a unitary group or orthogonal...

Elementary relative tractor calculus for Legendrean contact structures

Michał Andrzej Wasilewicz (2022)

Archivum Mathematicum

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For a manifold $M$ endowed with a Legendrean (or Lagrangean) contact structure $E \oplus F \subset T M$ , we give an elementary construction of an invariant partial connection on the quotient bundle $T M / F$ . This permits us to develop a naïve version of relative tractor calculus and to construct a second order invariant differential operator, which turns out to be the first relative BGG operator induced by the partial connection.

An obstruction to $ℓ^{p}$ -dimension

Nicolas Monod, Henrik Densing Petersen (2014)

Annales de l’institut Fourier

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Let $G$ be any group containing an infinite elementary amenable subgroup and let $2 < p < \infty$ . We construct an exhaustion of $ℓ^{p} G$ by closed invariant subspaces which all intersect trivially a fixed non-trivial closed invariant subspace. This is an obstacle to $ℓ^{p}$ -dimension and gives an answer to a question of Gaboriau.

Erratum to “Geometric rigidity of $\times m$ invariant measures” (J. Eur. Math. Soc. 14, 1539–1563 (2012))

Michael Hochman (2013)

Journal of the European Mathematical Society

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Limits of relatively hyperbolic groups and Lyndon’s completions

Olga Kharlampovich, Alexei Myasnikov (2012)

Journal of the European Mathematical Society

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We describe finitely generated groups $H$ universally equivalent (with constants from $G$ in the language) to a given torsion-free relatively hyperbolic group $G$ with free abelian parabolics. It turns out that, as in the free group case, the group $H$ embeds into the Lyndon’s completion $G^{ℤ [t]}$ of the group $G$ , or, equivalently, $H$ embeds into a group obtained from $G$ by finitely many extensions of centralizers. Conversely, every subgroup of $G^{ℤ [t]}$ containing $G$ is universally equivalent to $G$ . Since finitely...

Translation invariant forms on $L^{p} (G) (1 < p < \infty)$

Jean Bourgain (1986)

Annales de l'institut Fourier

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It is shown that if $G$ is a connected metrizable compact Abelian group and $1 < p < \infty$ , any (possibly discontinuous) translation invariant linear form on $L^{p} (G)$ is a scalar multiple of the Haar measure. This result extends the theorem of G.H. Meisters and W.M. Schmidt (J. Funct. Anal. 13 (1972), 407-424) on $L^{2} (G)$ . Our method permits in fact to consider any superreflexive translation invariant Banach lattice on $G$ , which is the adopted point of view. We study the representation of an element $f$ of this invariant...

The density of representation degrees

Martin Liebeck, Dan Segal, Aner Shalev (2012)

Journal of the European Mathematical Society

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For a group $G$ and a positive real number $x$ , define $d_{G} (x)$ to be the number of integers less than $x$ which are dimensions of irreducible complex representations of $G$ . We study the asymptotics of $d_{G} (x)$ for algebraic groups, arithmetic groups and finitely generated linear groups. In particular we prove an “alternative” for finitely generated linear groups $G$ in characteristic zero, showing that either there exists $α > 0$ such that $d_{G} (x) > x^{α}$ for all large $x$ , or $G$ is virtually abelian (in which case $d_{G} (x)$ is bounded). ...

Continuity of halo functions associated to homothecy invariant density bases

Oleksandra Beznosova, Paul Hagelstein (2014)

Colloquium Mathematicae

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Let be a collection of bounded open sets in ℝⁿ such that, for any x ∈ ℝⁿ, there exists a set U ∈ of arbitrarily small diameter containing x. The collection is said to be a density basis provided that, given a measurable set A ⊂ ℝⁿ, for a.e. x ∈ ℝⁿ we have $l i m_{k \to \infty} 1 / | R_{k} | \int_{R_{k}} χ_{A} = χ_{A} (x)$ for any sequence $R_{k}$ of sets in containing x whose diameters tend to 0. The geometric maximal operator $M_{}$ associated to is defined on L¹(ℝⁿ) by $M_{} f (x) = s u p_{x \in R \in} 1 / | R | \int_{R} | f |$ . The halo function ϕ of is defined on (1,∞) by $ϕ (u) = s u p 1 / | A | | x \in ℝ ⁿ : M_{} χ_{A} (x) > 1 / u | : 0 < | A | < \infty$ and on [0,1] by ϕ(u) = u. It is shown...

Invariant theory and the $𝒲_{1 + \infty}$ algebra with negative integral central charge

Andrew Linshaw (2011)

Journal of the European Mathematical Society

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The vertex algebra $𝒲_{1 + \infty, c}$ with central charge $c$ may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer $n \geq 1$ , it was conjectured in the physics literature that $𝒲_{1 + \infty, - n}$ should have a minimal strong generating set consisting of $n^{2} + 2 n$ elements. Using a free field realization of $𝒲_{1 + \infty, - n}$ due to Kac–Radul, together with a deformed version of Weyl’s first and second fundamental theorems of invariant theory for the standard representation...

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)...

Relations between Shy Sets and Sets of $ν_{p}$ -Measure Zero in Solovay’s Model

G. Pantsulaia (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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An example of a non-zero non-atomic translation-invariant Borel measure $ν_{p}$ on the Banach space $ℓ_{p} (1 \leq p \leq \infty)$ is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition " $ν_{p}$ -almost every element of $ℓ_{p}$ has a property P" implies that “almost every” element of $ℓ_{p}$ (in the sense of [4]) has the property P. It is also shown that the converse is not valid.

A new invariant and parametric connected sum of embeddings

A. Skopenkov (2007)

Fundamenta Mathematicae

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We define an isotopy invariant of embeddings $N \to ℝ^{m}$ of manifolds into Euclidean space. This invariant together with the α-invariant of Haefliger-Wu is complete in the dimension range where the α-invariant could be incomplete. We also define parametric connected sum of certain embeddings (analogous to surgery). This allows us to obtain new completeness results for the α-invariant and the following estimation of isotopy classes of embeddings. In the piecewise-linear category, for a (3n-2m+2)-connected...