On spectrality of the algebra of convolution dominated operators

Gero Fendle; Karlheinz Gröchenig; Michael Leinert

Displaying similar documents to “On spectrality of the algebra of convolution dominated operators”

Compact convolution operators between $L_{p} (G)$ -spaces

G. Crombez, W. Govaerts (1978)

Colloquium Mathematicae

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Operators which commute with convolutions on subspaces of $L_{\infty} (G)$

Anthony To-Ming Lau (1978)

Colloquium Mathematicae

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Translation-invariant operators on Lorentz spaces L(1,q) with 0 < q < 1

Leonardo Colzani, Peter Sjögren (1999)

Studia Mathematica

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We study convolution operators bounded on the non-normable Lorentz spaces $L^{1, q}$ of the real line and the torus. Here 0 < q < 1. On the real line, such an operator is given by convolution with a discrete measure, but on the torus a convolutor can also be an integrable function. We then give some necessary and some sufficient conditions for a measure or a function to be a convolutor on $L^{1, q}$ . In particular, when the positions of the atoms of a discrete measure are linearly independent over...

Convolution-dominated integral operators

Gero Fendler, Karlheinz Gröchenig, Michael Leinert (2010)

Banach Center Publications

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For a locally compact group G we consider the algebra CD(G) of convolution-dominated operators on L²(G), where an operator A: L²(G) → L²(G) is called convolution-dominated if there exists a ∈ L¹(G) such that for all f ∈ L²(G) |Af(x)| ≤ a⋆|f|(x), for almost all x ∈ G. (1) The case of discrete groups was treated in previous publications [, ]. For non-discrete groups we investigate a subalgebra of regular convolution-dominated operators generated by product convolution operators, where...

A comparison on the commutative neutrix convolution of distributions and the exchange formula

Adem Kiliçman (2001)

Czechoslovak Mathematical Journal

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Let $\tilde{f}$ , $\tilde{g}$ be ultradistributions in $𝒵^{'}$ and let ${\tilde{f}}_{n} = \tilde{f} * δ_{n}$ and ${\tilde{g}}_{n} = \tilde{g} * σ_{n}$ where ${δ_{n}}$ is a sequence in $𝒵$ which converges to the Dirac-delta function $δ$ . Then the neutrix product $\tilde{f} ⋄ \tilde{g}$ is defined on the space of ultradistributions $𝒵^{'}$ as the neutrix limit of the sequence ${\frac{1}{2} ({\tilde{f}}_{n} \tilde{g} + \tilde{f} {\tilde{g}}_{n})}$ provided the limit $\tilde{h}$ exist in the sense that $\underset{n \to \infty}{N - l i m} \frac{1}{2} 〈 {\tilde{f}}_{n} \tilde{g} + \tilde{f} {\tilde{g}}_{n}, ψ 〉 = 〈 \tilde{h}, ψ 〉$ for all $ψ$ in $𝒵$ . We also prove that the neutrix convolution product $f ♦ * g$ exist in $𝒟^{'}$ , if and only if the neutrix product $\tilde{f} ⋄ \tilde{g}$ exist in $𝒵^{'}$ and the exchange formula $F (f ♦ * g) = \tilde{f} ⋄ \tilde{g}$ is then satisfied.

$L^{p} (ℝ ⁿ)$ bounds for commutators of convolution operators

Guoen Hu, Qiyu Sun, Xin Wang (2002)

Colloquium Mathematicae

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The $L^{p} (ℝ ⁿ)$ boundedness is established for commutators generated by BMO(ℝⁿ) functions and convolution operators whose kernels satisfy certain Fourier transform estimates. As an application, a new result about the $L^{p} (ℝ ⁿ)$ boundedness is obtained for commutators of homogeneous singular integral operators whose kernels satisfy the Grafakos-Stefanov condition.

The class of convolution operators on the Marcinkiewicz spaces

Ka-Sing Lau (1981)

Annales de l'institut Fourier

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Let $𝒯_{X}$ denote the operator-norm closure of the class of convolution operators $Φ_{μ} : X \to X$ where $X$ is a suitable function space on $R$ . Let $ℳ_{r}^{p}$ be the closed subspace of regular functions in the Marinkiewicz space $ℳ^{p}$ , $1 \leq p < \infty$ . We show that the space $𝒯_{ℳ_{r}^{p}}$ is isometrically isomorphic to $𝒯_{L^{p}}$ and that strong operator sequential convergence and norm convergence in $𝒯_{ℳ_{r}^{p}}$ coincide. We also obtain some results concerning convolution operators under the Wiener transformation. These are to improve a Tauberian theorem of Wiener...

Characterization of the convolution operators on quasianalytic classes of Beurling type that admit a continuous linear right inverse

José Bonet, Reinhold Meise (2008)

Studia Mathematica

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Extending previous work by Meise and Vogt, we characterize those convolution operators, defined on the space $_{(ω)} (ℝ)$ of (ω)-quasianalytic functions of Beurling type of one variable, which admit a continuous linear right inverse. Also, we characterize those (ω)-ultradifferential operators which admit a continuous linear right inverse on $_{(ω)} [a, b]$ for each compact interval [a,b] and we show that this property is in fact weaker than the existence of a continuous linear right inverse on $_{(ω)} (ℝ)$ .

The $V_{a}$ -deformation of the classical convolution

Anna Dorota Krystek (2007)

Banach Center Publications

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We study deformations of the classical convolution. For every invertible transformation T:μ ↦ Tμ, we are able to define a new associative convolution of measures by $μ *_{T} ν = T^{- 1} (T μ * T ν)$ . We deal with the $V_{a}$ -deformation of the classical convolution. We prove the analogue of the classical Lévy-Khintchine formula. We also show the central limit measure, which turns out to be the standard Gaussian measure. Moreover, we calculate the Poisson measure in the $V_{a}$ -deformed classical convolution and give the orthogonal...

On the product formula on noncompact Grassmannians

Piotr Graczyk, Patrice Sawyer (2013)

Colloquium Mathematicae

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We study the absolute continuity of the convolution $δ_{e^{X}}^{♮} * δ_{e^{Y}}^{♮}$ of two orbital measures on the symmetric space SO₀(p,q)/SO(p)×SO(q), q > p. We prove sharp conditions on X,Y ∈ for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions. We show that the sharp criterion developed for SO₀(p,q)/SO(p)×SO(q) also serves for the spaces SU(p,q)/S(U(p)×U(q)) and Sp(p,q)/Sp(p)×Sp(q), q > p. We moreover apply our results to...

Hankel determinant for a class of analytic functions of complex order defined by convolution

S. M. El-Deeb, M. K. Aouf (2015)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper, we obtain the Fekete-Szego inequalities for the functions of complex order defined by convolution. Also, we find upper bounds for the second Hankel determinant $| a_{2} a_{4} - a_{3}^{2} |$ for functions belonging to the class $S_{γ}^{b} (g (z); A, B)$ .

A convolution property of the Cantor-Lebesgue measure, II

Daniel M. Oberlin (2003)

Colloquium Mathematicae

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For 1 ≤ p,q ≤ ∞, we prove that the convolution operator generated by the Cantor-Lebesgue measure on the circle is a contraction whenever it is bounded from $L^{p} ()$ to $L^{q} ()$ . We also give a condition on p which is necessary if this operator maps $L^{p} ()$ into L²().

Distributional {D}unkl transform and {D}unkl convolution operators

Jorge J. Betancor (2006)

Bollettino dell'Unione Matematica Italiana

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In this paper, that is divided in two parts, we study the distributional Dunkl transform on R. In the first part we investigate the Dunkl transform and the Dunkl convolution operators on tempered distributions. We prove that the tempered distributions defining Dunkl convolution operators on the Schwartz space are the elements of $\mathcal{O}^{\prime}_{c}$ , the space of usual convolution operators on $S$ . In the second part we define the distributional Dunkl transform by employing the kernel method. We introduce...

One-parameter semigroups in the convolution algebra of rapidly decreasing distributions

(2012)

Colloquium Mathematicae

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The paper is devoted to infinitely differentiable one-parameter convolution semigroups in the convolution algebra $_{C}^{'} (ℝ ⁿ; M_{m \times m})$ of matrix valued rapidly decreasing distributions on ℝⁿ. It is proved that $G \in_{C}^{'} (ℝ ⁿ; M_{m \times m})$ is the generating distribution of an i.d.c.s. if and only if the operator $\partial_{t} \otimes_{m \times m} - G *$ on $ℝ^{1 + n}$ satisfies the Petrovskiĭ condition for forward evolution. Some consequences are discussed.

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...

Standard ideals in convolution Sobolev algebras on the half-line

José E. Galé, Antoni Wawrzyńczyk (2011)

Colloquium Mathematicae

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We study the relation between standard ideals of the convolution Sobolev algebra $₊^{(n)} (t ⁿ)$ and the convolution Beurling algebra L¹((1+t)ⁿ) on the half-line (0,∞). In particular it is proved that all closed ideals in $₊^{(n)} (t ⁿ)$ with compact and countable hull are standard.

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...

The type set for some measures on $ℝ^{2 n}$ with $n$ -dimensional support

E. Ferreyra, T. Godoy, Marta Urciuolo (2002)

Czechoslovak Mathematical Journal

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Let $ϕ_{1}, \dots, ϕ_{n}$ be real homogeneous functions in $C^{\infty} (ℝ^{n} - {0})$ of degree $k \geq 2$ , let $ϕ (x) = (ϕ_{1} (x), \dots, ϕ_{n} (x))$ and let $μ$ be the Borel measure on $ℝ^{2 n}$ given by $μ (E) = \int_{ℝ^{n}} χ_{E} (x, ϕ (x)) {| x |}^{γ - n} d x$ where $d x$ denotes the Lebesgue measure on $ℝ^{n}$ and $γ > 0$ . Let $T_{μ}$ be the convolution operator $T_{μ} f (x) = (μ * f) (x)$ and let $E_{μ} = {(1 / p, 1 / q) ∥ T_{μ} ∥_{p, q} < \infty, 1 \leq p, q \leq \infty} .$ Assume that, for $x \neq 0$ , the following two conditions hold: $det (d^{2} ϕ (x) h)$ vanishes only at $h = 0$ and $det (d ϕ (x)) \neq 0$ . In this paper we show that if $γ > n (k + 1) / 3$ then $E_{μ}$ is the empty set and if $γ \leq n (k + 1) / 3$ then $E_{μ}$ is the closed segment with endpoints $D = (1 - \frac{γ}{n (k + 1)}, 1 - \frac{2 γ}{n (k + 1)})$ and $D^{'} = (\frac{2 γ}{n (1 + k)}, \frac{γ}{n (1 + k)})$ . Also, we give some examples.

Absolutely continuous linear operators on Köthe-Bochner spaces

(2011)

Banach Center Publications

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Let E be a Banach function space over a finite and atomless measure space (Ω,Σ,μ) and let $(X, | | \cdot | |_{X})$ and $(Y, | | \cdot | |_{Y})$ be real Banach spaces. A linear operator T acting from the Köthe-Bochner space E(X) to Y is said to be absolutely continuous if $| | T (1_{A ₙ} f) {| |}_{Y} \to 0$ whenever μ(Aₙ) → 0, (Aₙ) ⊂ Σ. In this paper we examine absolutely continuous operators from E(X) to Y. Moreover, we establish relationships between different classes of linear operators from E(X) to Y.