A finite multiplicity Helson-Lowdenslager-de Branges theorem

Sneh Lata; Meghna Mittal; Dinesh Singh

Displaying similar documents to “A finite multiplicity Helson-Lowdenslager-de Branges theorem”

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

W. Chojnacki (1978)

Colloquium Mathematicae

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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...

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.

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.

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...

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.

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.

The Daugavet property and translation-invariant subspaces

Simon Lücking (2014)

Studia Mathematica

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Let G be an infinite, compact abelian group and let Λ be a subset of its dual group Γ. We study the question which spaces of the form $C_{Λ} (G)$ or $L ¹_{Λ} (G)$ and which quotients of the form $C (G) / C_{Λ} (G)$ or $L ¹ (G) / L ¹_{Λ} (G)$ have the Daugavet property. We show that $C_{Λ} (G)$ is a rich subspace of C(G) if and only if $Γ ∖ Λ^{- 1}$ is a semi-Riesz set. If $L ¹_{Λ} (G)$ is a rich subspace of L¹(G), then $C_{Λ} (G)$ is a rich subspace of C(G) as well. Concerning quotients, we prove that $C (G) / C_{Λ} (G)$ has the Daugavet property if Λ is a Rosenthal set, and that $L ¹_{Λ} (G)$ is a poor subspace of L¹(G)...

Separating by $G_{δ}$ -sets in finite powers of ω₁

Yasushi Hirata, Nobuyuki Kemoto (2003)

Fundamenta Mathematicae

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It is known that all subspaces of ω₁² have the property that every pair of disjoint closed sets can be separated by disjoint $G_{δ}$ -sets (see [4]). It has been conjectured that all subspaces of ω₁ⁿ also have this property for each n < ω. We exhibit a subspace of ⟨α,β,γ⟩ ∈ ω₁³: α ≤ β ≤ γ which does not have this property, thus disproving the conjecture. On the other hand, we prove that all subspaces of ⟨α,β,γ⟩ ∈ ω₁³: α < β < γ have this property.

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...

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...

Invariant subspaces for operators in a general II1-factor

Uffe Haagerup, Hanne Schultz (2009)

Publications Mathématiques de l'IHÉS

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Let ℳ be a von Neumann factor of type II1 with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace $𝒦 = 𝒦_{T} (B)$ affiliated with ℳ, such that the Brown measure of ${T |}_{𝒦}$ is concentrated...

Completely 1-complemented subspaces of the Schatten spaces $S^{p}$

Jean Roydor (2007)

Banach Center Publications

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We describe the subspaces of $S^{p}$ (1 ≤ p ≠ 2 < ∞) which are the range of a completely contractive projection.

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...

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...

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.

Structure of Rademacher subspaces in Cesàro type spaces

Sergey V. Astashkin, Lech Maligranda (2015)

Studia Mathematica

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The structure of the closed linear span of the Rademacher functions in the Cesàro space $C e s_{\infty}$ is investigated. It is shown that every infinite-dimensional subspace of either is isomorphic to l₂ and uncomplemented in $C e s_{\infty}$ , or contains a subspace isomorphic to c₀ and complemented in . The situation is rather different in the p-convexification of $C e s_{\infty}$ if 1 < p < ∞.

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...

Khinchin inequality and Banach-Saks type properties in rearrangement-invariant spaces

F. A. Sukochev, D. Zanin (2009)

Studia Mathematica

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We study the class of all rearrangement-invariant ( = r.i.) function spaces E on [0,1] such that there exists 0 < q < 1 for which $∥ \sum_{k = 1}^{n} ξ_{k} ∥_{E} \leq C n^{q}$ , where ${ξ_{k}}_{k \geq 1} \subset E$ is an arbitrary sequence of independent identically distributed symmetric random variables on [0,1] and C > 0 does not depend on n. We completely characterize all Lorentz spaces having this property and complement classical results of Rodin and Semenov for Orlicz spaces $e x p (L_{p})$ , p ≥ 1. We further apply our results to the study of Banach-Saks...

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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On an Invariant Borel Measure in Hilbert Space

G. Pantsulaia (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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An example of a nonzero σ-finite Borel measure μ with everywhere dense linear manifold $_{μ}$ of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space ℓ₂ such that μ and any shift $μ^{(a)}$ of μ by a vector $a \in ℓ ₂ ∖_{μ}$ are neither equivalent nor orthogonal. This extends a result established in [7].

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...

On $C_{0 \cdot}$ multi-contractions having a regular dilation

Dan Popovici (2005)

Studia Mathematica

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Commuting multi-contractions of class $C_{0 \cdot}$ and having a regular isometric dilation are studied. We prove that a polydisc contraction of class $C_{0 \cdot}$ is the restriction of a backwards multi-shift to an invariant subspace, extending a particular case of a result by R. E. Curto and F.-H. Vasilescu. A new condition on a commuting multi-operator, which is equivalent to the existence of a regular isometric dilation and improves a recent result of A. Olofsson, is obtained as a consequence. ...

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,...

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.