Addition to the paper “Translation invariant subspaces of $L_{p}(G)$” Studia Mathematica 48 (1973), pp. 245-250

Aharon Atzmon

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Shift-modulation invariant spaces on LCA groups

Carlos Cabrelli, Victoria Paternostro (2012)

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A (K,Λ) shift-modulation invariant space is a subspace of L²(G) that is invariant under translations along elements in K and modulations by elements in Λ. Here G is a locally compact abelian group, and K and Λ are closed subgroups of G and the dual group Ĝ, respectively. We provide a characterization of shift-modulation invariant spaces when K and Λ are uniform lattices. This extends previous results known for $L ² (ℝ^{d})$ . We develop fiberization techniques and suitable range functions adapted...

Linear combinations of generators in multiplicatively invariant spaces

Victoria Paternostro (2015)

Studia Mathematica

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Multiplicatively invariant (MI) spaces are closed subspaces of L²(Ω, ) that are invariant under multiplication by (some) functions in $L^{\infty} (Ω)$ ; they were first introduced by Bownik and Ross (2014). In this paper we work with MI spaces that are finitely generated. We prove that almost every set of functions constructed by taking linear combinations of the generators of a finitely generated MI space is a new set of generators for the same space, and we give necessary and sufficient conditions...

Infinite dimensional linear groups with a large family of $G$ -invariant subspaces

L. A. Kurdachenko, A. V. Sadovnichenko, I. Ya. Subbotin (2010)

Commentationes Mathematicae Universitatis Carolinae

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Let $F$ be a field, $A$ be a vector space over $F$ , $GL (F, A)$ be the group of all automorphisms of the vector space $A$ . A subspace $B$ is called almost $G$ -invariant, if ${dim}_{F} (B / {Core}_{G} (B))$ is finite. In the current article, we begin the study of those subgroups $G$ of $GL (F, A)$ for which every subspace of $A$ is almost $G$ -invariant. More precisely, we consider the case when $G$ is a periodic group. We prove that in this case $A$ includes a $G$ -invariant subspace $B$ of finite codimension whose subspaces are $G$ -invariant.

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...

On projections of $L^{\infty} (G)$ onto translation-invariant subspaces

W. Chojnacki (1978)

Colloquium Mathematicae

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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...

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...

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...

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...

Strictly singular inclusions of rearrangement invariant spaces and Rademacher spaces

Sergei V. Astashkin, Francisco L. Hernández, Evgeni M. Semenov (2009)

Studia Mathematica

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If G is the closure of $L_{\infty}$ in exp L₂, it is proved that the inclusion between rearrangement invariant spaces E ⊂ F is strictly singular if and only if it is disjointly strictly singular and E ⊊ G. For any Marcinkiewicz space M(φ) ⊂ G such that M(φ) is not an interpolation space between $L_{\infty}$ and G it is proved that there exists another Marcinkiewicz space M(ψ) ⊊ M(φ) with the property that the M(ψ) and M(φ) norms are equivalent on the Rademacher subspace. Applications are given and a question...

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...

Composition operators: $N_{α}$ to the Bloch space to $Q_{β}$

Jie Xiao (2000)

Studia Mathematica

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Let $N_{α}$ ,B and Qβ be the weighted Nevanlinna space, the Bloch space and the Q space, respectively. Note that B and $Q_{β}$ are Möbius invariant, but $N_{α}$ is not. We characterize, in function-theoretic terms, when the composition operator $C_{ϕ} f = f ◦ ϕ$ induced by an analytic self-map ϕ of the unit disk defines an operator $C_{ϕ} : N_{α} \to B$ , $B \to Q_{β}$ , $N_{α} \to Q_{β}$ which is bounded resp. compact.

Isomorphic classification of the tensor products $E ₀ (e x p α i) \otimes ̂ E_{\infty} (e x p β j)$

Peter Chalov, Vyacheslav Zakharyuta (2011)

Studia Mathematica

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It is proved, using so-called multirectangular invariants, that the condition αβ = α̃β̃ is sufficient for the isomorphism of the spaces $E ₀ (e x p α i) \otimes ̂ E_{\infty} (e x p β j)$ and $E ₀ (e x p α ̃ i) \otimes ̂ E_{\infty} (e x p β ̃ j)$ . This solves a problem posed in [14, 15, 1]. Notice that the necessity has been proved earlier in [14].

A characterization of the minimal strongly character invariant Segal algebra

Viktor Losert (1980)

Annales de l'institut Fourier

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For a locally compact, abelian group $G$ , we study the space $S_{0} (G)$ of functions on $G$ belonging locally to the Fourier algebra and with $l^{1}$ -behavior at infinity. We give an abstract characterization of the family of spaces ${S_{0} (G) : G$ abelian $}$ by its hereditary properties.

Displaying similar documents to “Addition to the paper “Translation invariant subspaces of L p ( G ) ” Studia Mathematica 48 (1973), pp. 245-250”

Displaying similar documents to “Addition to the paper “Translation invariant subspaces of $L_{p} (G)$ ” Studia Mathematica 48 (1973), pp. 245-250”