Extending Maps in Hilbert Manifolds

Piotr Niemiec

Displaying similar documents to “Extending Maps in Hilbert Manifolds”

On the separation properties of $K_{ω}$

Malgorzata Wójcicka (1986)

Colloquium Mathematicae

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On isometrical extension properties of function spaces

Hisao Kato (2015)

Commentationes Mathematicae Universitatis Carolinae

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In this note, we prove that any “bounded” isometries of separable metric spaces can be represented as restrictions of linear isometries of function spaces $C (Q)$ and $C (Δ)$ , where $Q$ and $Δ$ denote the Hilbert cube ${[0, 1]}^{\infty}$ and a Cantor set, respectively.

Operators on a Hilbert space similar to a part of the backward shift of multiplicity one

Yoichi Uetake (2001)

Studia Mathematica

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Let A: X → X be a bounded operator on a separable complex Hilbert space X with an inner product $⟨ \cdot, \cdot ⟩_{X}$ . For b, c ∈ X, a weak resolvent of A is the complex function of the form $⟨ {(I - z A)}^{- 1} b, c ⟩_{X}$ . We will discuss an equivalent condition, in terms of weak resolvents, for A to be similar to a restriction of the backward shift of multiplicity 1.

On the classification of $3$ -dimensional $F$ -manifold algebras

Zhiqi Chen, Jifu Li, Ming Ding (2022)

Czechoslovak Mathematical Journal

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$F$ -manifold algebras are focused on the algebraic properties of the tangent sheaf of $F$ -manifolds. The local classification of 3-dimensional $F$ -manifolds has been given in A. Basalaev, C. Hertling (2021). We study the classification of 3-dimensional $F$ -manifold algebras over the complex field $ℂ$ .

Some equivalent metrics for bounded normal operators

Mohammad Reza Jabbarzadeh, Rana Hajipouri (2018)

Mathematica Bohemica

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Some stronger and equivalent metrics are defined on $ℳ$ , the set of all bounded normal operators on a Hilbert space $H$ and then some topological properties of $ℳ$ are investigated.

Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

Sunanda Naik, Karabi Rajbangshi (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let f be an analytic function on the unit disk . We define a generalized Hilbert-type operator $_{a, b}$ by $_{a, b} (f) (z) = Γ (a + 1) / Γ (b + 1) \int_{0}^{1} (f (t) {(1 - t)}^{b}) / ({(1 - t z)}^{a + 1}) d t$ , where a and b are non-negative real numbers. In particular, for a = b = β, $_{a, b}$ becomes the generalized Hilbert operator $_{β}$ , and β = 0 gives the classical Hilbert operator . In this article, we find conditions on a and b such that $_{a, b}$ is bounded on Dirichlet-type spaces $S^{p}$ , 0 < p < 2, and on Bergman spaces $A^{p}$ , 2 < p < ∞. Also we find an upper bound for the norm of the operator $_{a, b}$ ....

Hilbert series of the Grassmannian and $k$ -Narayana numbers

Lukas Braun (2019)

Communications in Mathematics

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We compute the Hilbert series of the complex Grassmannian using invariant theoretic methods. This is made possible by showing that the denominator of the $q$ -Hilbert series is a Vandermonde-like determinant. We show that the $h$ -polynomial of the Grassmannian coincides with the $k$ -Narayana polynomial. A simplified formula for the $h$ -polynomial of Schubert varieties is given. Finally, we use a generalized hypergeometric Euler transform to find simplified formulae for the $k$ -Narayana numbers,...

Hyperspaces of Finite Sets in Universal Spaces for Absolute Borel Classes

Kotaro Mine, Katsuro Sakai, Masato Yaguchi (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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By Fin(X) (resp. $F i n^{k} (X)$ ), we denote the hyperspace of all non-empty finite subsets of X (resp. consisting of at most k points) with the Vietoris topology. Let ℓ₂(τ) be the Hilbert space with weight τ and $ℓ ₂^{f} (τ)$ the linear span of the canonical orthonormal basis of ℓ₂(τ). It is shown that if $E = ℓ ₂^{f} (τ)$ or E is an absorbing set in ℓ₂(τ) for one of the absolute Borel classes $_{α} (τ)$ and $_{α} (τ)$ of weight ≤ τ (α > 0) then Fin(E) and each $F i n^{k} (E)$ are homeomorphic to E. More generally, if X is a connected E-manifold then Fin(X)...

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].

Exotic Deformations of Calabi-Yau Manifolds

Paolo de Bartolomeis, Adriano Tomassini (2013)

Annales de l’institut Fourier

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We introduce Quantum Inner State manifolds (QIS manifolds) as (compact) $2 n$ -dimensional symplectic manifolds $(M, κ)$ endowed with a $κ$ -tamed almost complex structure $J$ and with a nowhere vanishing and normalized section $ϵ$ of the bundle $Λ_{J}^{n, 0} (M)$ satisfying the condition ${\bar{\partial}}_{J} ϵ = 0$ . We study the moduli space $𝔐$ of QIS deformations of a given Calabi-Yau manifold, computing its tangent space...

On varieties of Hilbert type

Lior Bary-Soroker, Arno Fehm, Sebastian Petersen (2014)

Annales de l’institut Fourier

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A variety $X$ over a field $K$ is of Hilbert type if $X (K)$ is not thin. We prove that if $f : X \to S$ is a dominant morphism of $K$ -varieties and both $S$ and all fibers $f^{- 1} (s)$ , $s \in S (K)$ , are of Hilbert type, then so is $X$ . We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Thélène and Sansuc on algebraic groups.

On coarse embeddability into $ℓ_{p}$ -spaces and a conjecture of Dranishnikov

Piotr W. Nowak (2006)

Fundamenta Mathematicae

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We show that the Hilbert space is coarsely embeddable into any $ℓ_{p}$ for 1 ≤ p ≤ ∞. It follows that coarse embeddability into ℓ₂ and into $ℓ_{p}$ are equivalent for 1 ≤ p < 2.

Failure of the Factor Theorem for Borel pre-Hilbert spaces

Tadeusz Dobrowolski, Witold Marciszewski (2002)

Fundamenta Mathematicae

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In every infinite-dimensional Fréchet space X, we construct a linear subspace E such that E is an $F_{σ δ σ}$ -subset of X and contains a retract R so that $R \times E^{ω}$ is not homeomorphic to $E^{ω}$ . This shows that Toruńczyk’s Factor Theorem fails in the Borel case.

On the Kaczmarz algorithm of approximation in infinite-dimensional spaces

Stanisław Kwapień, Jan Mycielski (2001)

Studia Mathematica

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The Kaczmarz algorithm of successive projections suggests the following concept. A sequence $(e_{k})$ of unit vectors in a Hilbert space is said to be effective if for each vector x in the space the sequence (xₙ) converges to x where (xₙ) is defined inductively: x₀ = 0 and $x ₙ = x_{n - 1} + α ₙ e ₙ$ , where $α ₙ = ⟨ x - x_{n - 1}, e ₙ ⟩$ . We prove the effectivity of some sequences in Hilbert spaces. We generalize the concept of effectivity to sequences of vectors in Banach spaces and we prove some results for this more general concept.

Homeomorphism groups of Sierpiński carpets and Erdős space

Jan J. Dijkstra, Dave Visser (2010)

Fundamenta Mathematicae

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Erdős space is the “rational” Hilbert space, that is, the set of vectors in ℓ² with all coordinates rational. Erdős proved that is one-dimensional and homeomorphic to its own square × , which makes it an important example in dimension theory. Dijkstra and van Mill found topological characterizations of . Let $M ₙ^{n + 1}$ , n ∈ ℕ, be the n-dimensional Menger continuum in $ℝ^{n + 1}$ , also known as the n-dimensional Sierpiński carpet, and let D be a countable dense subset of $M ₙ^{n + 1}$ . We consider the topological group...

$Z ₂^{k}$ -actions with a special fixed point set

Pedro L. Q. Pergher, Rogério de Oliveira (2005)

Fundamenta Mathematicae

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Let Fⁿ be a connected, smooth and closed n-dimensional manifold satisfying the following property: if $N^{m}$ is any smooth and closed m-dimensional manifold with m > n and $T : N^{m} \to N^{m}$ is a smooth involution whose fixed point set is Fⁿ, then m = 2n. We describe the equivariant cobordism classification of smooth actions $(M^{m}; Φ)$ of the group $G = Z ₂^{k}$ on closed smooth m-dimensional manifolds $M^{m}$ for which the fixed point set of the action is a submanifold Fⁿ with the above property. This generalizes a result of F....

The Killing Tensors on an $n$ -dimensional Manifold with $S L (n,)$ -structure

Sergey E. Stepanov, Irina I. Tsyganok, Marina B. Khripunova (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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In this paper we solve the problem of finding integrals of equations determining the Killing tensors on an $n$ -dimensional differentiable manifold $M$ endowed with an equiaffine $S L (n,)$ -structure and discuss possible applications of obtained results in Riemannian geometry.

On the Hilbert $2$ -class field tower of some imaginary biquadratic number fields

Mohamed Mahmoud Chems-Eddin, Abdelmalek Azizi, Abdelkader Zekhnini, Idriss Jerrari (2021)

Czechoslovak Mathematical Journal

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Let $𝕜 = ℚ (\sqrt{2}, \sqrt{d})$ be an imaginary bicyclic biquadratic number field, where $d$ is an odd negative square-free integer and $𝕜_{2}^{(2)}$ its second Hilbert $2$ -class field. Denote by $G = Gal (𝕜_{2}^{(2)} / 𝕜)$ the Galois group of $𝕜_{2}^{(2)} / 𝕜$ . The purpose of this note is to investigate the Hilbert $2$ -class field tower of $𝕜$ and then deduce the structure of $G$ .