Equivalences involving (p,q)-multi-norms

Oscar Blasco; H. G. Dales; Hung Le Pham

Displaying similar documents to “Equivalences involving (p,q)-multi-norms”

Equivalence of multi-norms

H. G. Dales, M. Daws, H. L. Pham, P. Ramsden

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The theory of multi-norms was developed by H. G. Dales and M. E. Polyakov in a memoir that was published in Dissertationes Mathematicae. In that memoir, the notion of ’equivalence’ of multi-norms was defined. In the present memoir, we make a systematic study of when various pairs of multi-norms are mutually equivalent. In particular, we study when (p,q)-multi-norms defined on spaces $L^{r} (Ω)$ are equivalent, resolving most cases; we have stronger results in the case where r = 2. We also show...

Absolutely continuous linear operators on Köthe-Bochner spaces

(2011)

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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.

Stability of infinite ranges and kernels

K.-H. Förster, V. Müller (2006)

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Let A(·) be a regular function defined on a connected metric space G whose values are mutually commuting essentially Kato operators in a Banach space. Then the spaces $R^{\infty} (A (z))$ and $\bar{N^{\infty} (A (z))}$ do not depend on z ∈ G. This generalizes results of B. Aupetit and J. Zemánek.

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...

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Jan A. Rempała (1990)

Annales Polonici Mathematici

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Spaces of operators and c₀

P. Lewis (2001)

Studia Mathematica

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Bessaga and Pełczyński showed that if c₀ embeds in the dual X* of a Banach space X, then ℓ¹ embeds complementably in X, and $ℓ^{\infty}$ embeds as a subspace of X*. In this note the Diestel-Faires theorem and techniques of Kalton are used to show that if X is an infinite-dimensional Banach space, Y is an arbitrary Banach space, and c₀ embeds in L(X,Y), then $ℓ^{\infty}$ embeds in L(X,Y), and ℓ¹ embeds complementably in $X \otimes_{γ} Y *$ . Applications to embeddings of c₀ in various spaces of operators are given.

Spaces of compact operators on $C (2^{} \times [0, α])$ spaces

Elói Medina Galego (2011)

Colloquium Mathematicae

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We classify, up to isomorphism, the spaces of compact operators (E,F), where E and F are the Banach spaces of all continuous functions defined on the compact spaces $2^{} \times [0, α]$ , the topological products of Cantor cubes $2^{}$ and intervals of ordinal numbers [0,α].

Operator Lipschitz functions on Banach spaces

Jan Rozendaal, Fedor Sukochev, Anna Tomskova (2016)

Studia Mathematica

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Let X, Y be Banach spaces and let (X,Y) be the space of bounded linear operators from X to Y. We develop the theory of double operator integrals on (X,Y) and apply this theory to obtain commutator estimates of the form $| | f (B) S - {S f (A) | |}_{(X, Y)} \leq c o n s t | | B S - {S A | |}_{(X, Y)}$ for a large class of functions f, where A ∈ (X), B ∈ (Y) are scalar type operators and S ∈ (X,Y). In particular, we establish this estimate for f(t): = |t| and for diagonalizable operators on $X = ℓ_{p}$ and $Y = ℓ_{q}$ for p < q. We also study the estimate above in the setting of Banach...

The ideal of p-compact operators: a tensor product approach

Daniel Galicer, Silvia Lassalle, Pablo Turco (2012)

Studia Mathematica

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We study the space of p-compact operators, $_{p}$ , using the theory of tensor norms and operator ideals. We prove that $_{p}$ is associated to $/ d_{p}$ , the left injective associate of the Chevet-Saphar tensor norm $d_{p}$ (which is equal to $g_{p^{'}}^{'}$ ). This allows us to relate the theory of p-summing operators to that of p-compact operators. Using the results known for the former class and appropriate hypotheses on E and F we prove that $_{p} (E; F)$ is equal to $_{q} (E; F)$ for a wide range of values of p and q, and show that our results...

Some properties and applications of equicompact sets of operators

E. Serrano, C. Piñeiro, J. M. Delgado (2007)

Studia Mathematica

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Let X and Y be Banach spaces. A subset M of (X,Y) (the vector space of all compact operators from X into Y endowed with the operator norm) is said to be equicompact if every bounded sequence (xₙ) in X has a subsequence $(x_{k (n)}) ₙ$ such that $(T x_{k (n)}) ₙ$ is uniformly convergent for T ∈ M. We study the relationship between this concept and the notion of uniformly completely continuous set and give some applications. Among other results, we obtain a generalization of the classical Ascoli theorem and a compactness...

Isomorphic properties in spaces of compact operators

Ioana Ghenciu (2023)

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We introduce the definition of $p$ -limited completely continuous operators, $1 \leq p < \infty$ . The question of whether a space of operators has the property that every $p$ -limited subset is relative compact when the dual of the domain and the codomain have this property is studied using $p$ -limited completely continuous evaluation operators.

Regularity of domains of parameterized families of closed linear operators

Teresa Winiarska, Tadeusz Winiarski (2003)

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The purpose of this paper is to provide a method of reduction of some problems concerning families $A_{t} = {(A (t))}_{t \in}$ of linear operators with domains ${(_{t})}_{t \in}$ to a problem in which all the operators have the same domain . To do it we propose to construct a family ${(Ψ_{t})}_{t \in}$ of automorphisms of a given Banach space X having two properties: (i) the mapping $t \mapsto Ψ_{t}$ is sufficiently regular and (ii) $Ψ_{t} () =_{t}$ for t ∈ . Three effective constructions are presented: for elliptic operators of second order with the Robin boundary condition...

The Embeddability of c₀ in Spaces of Operators

Ioana Ghenciu, Paul Lewis (2008)

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Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space $K_{w *} (X *, Y)$ , as well as the complementation of the space $K_{w *} (X *, Y)$ of w*-w compact operators in the space $L_{w *} (X *, Y)$ of w*-w operators from X* to Y.

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...

Products of n open subsets in the space of continuous functions on [0,1]

Ehrhard Behrends (2011)

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Let O₁,...,Oₙ be open sets in C[0,1], the space of real-valued continuous functions on [0,1]. The product O₁ ⋯ Oₙ will in general not be open, and in order to understand when this can happen we study the following problem: given f₁,..., fₙ ∈ C[0,1], when is it true that f₁ ⋯ fₙ lies in the interior of $B_{ε} (f ₁) \dots B_{ε} (f ₙ)$ for all ε > 0 ? ( $B_{ε}$ denotes the closed ball with radius ε and centre f.) The main result of this paper is a characterization in terms of the walk t ↦ γ(t): = (f₁(t),..., fₙ(t)) in ℝⁿ....

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Deba P. Sinha, Anil K. Karn (2002)

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For p ≥ 1, a subset K of a Banach space X is said to be relatively p-compact if $K \subset \sum_{n = 1}^{\infty} α ₙ x ₙ : α ₙ \in B a l l (l_{p^{'}})$ , where p’ = p/(p-1) and $x ₙ \in l_{p}^{s} (X)$ . An operator T ∈ B(X,Y) is said to be p-compact if T(Ball(X)) is relatively p-compact in Y. Similarly, weak p-compactness may be defined by considering $x ₙ \in l_{p}^{w} (X)$ . It is proved that T is (weakly) p-compact if and only if T* factors through a subspace of $l_{p}$ in a particular manner. The normed operator ideals $(K_{p}, κ_{p})$ of p-compact operators and $(W_{p}, ω_{p})$ of weakly p-compact operators, arising from these factorizations,...

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On Lindenstrauss-Pełczyński spaces

Jesús M. F. Castillo, Yolanda Moreno, Jesús Suárez (2006)

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We consider some stability aspects of the classical problem of extension of C(K)-valued operators. We introduce the class ℒ of Banach spaces of Lindenstrauss-Pełczyński type as those such that every operator from a subspace of c₀ into them can be extended to c₀. We show that all ℒ-spaces are of type $ℒ_{\infty}$ but not conversely. Moreover, $ℒ_{\infty}$ -spaces will be characterized as those spaces E such that E-valued operators from w*(l₁,c₀)-closed subspaces of l₁ extend to l₁. Regarding examples we will...

Non-hyperreflexive reflexive spaces of operators

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The classical subspaces of the projective tensor products of $ℓ_{p}$ and C(α) spaces, α < ω₁

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We completely determine the $ℓ_{q}$ and C(K) spaces which are isomorphic to a subspace of $ℓ_{p} \otimes ̂_{π} C (α)$ , the projective tensor product of the classical $ℓ_{p}$ space, 1 ≤ p < ∞, and the space C(α) of all scalar valued continuous functions defined on the interval of ordinal numbers [1,α], α < ω₁. In order to do this, we extend a result of A. Tong concerning diagonal block matrices representing operators from $ℓ_{p}$ to ℓ₁, 1 ≤ p < ∞. The first main theorem is an extension of a result of E. Oja and states...

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Tensor product of left n-invertible operators

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A Banach space operator T ∈ has a left m-inverse (resp., an essential left m-inverse) for some integer m ≥ 1 if there exists an operator S ∈ (resp., an operator S ∈ and a compact operator K ∈ ) such that $\sum_{i = 0}^{m} {(- 1)}^{i} (\binom{m}{i}) S^{m - i} T^{m - i} = 0$ (resp., $\sum_{i = 0}^{m} {(- 1)}^{i} (\binom{m}{i}) T^{m - i} S^{m - i} = K$ ). If $T_{i}$ is left $m_{i}$ -invertible (resp., essentially left $m_{i}$ -invertible), then the tensor product T₁ ⊗ T₂ is left (m₁ + m₂-1)-invertible (resp., essentially left (m₁ + m₂-1)-invertible). Furthermore, if T₁ is strictly left m-invertible (resp., strictly essentially left m-invertible),...

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Order-bounded operators from vector-valued function spaces to Banach spaces

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Let E be an ideal of L⁰ over a σ-finite measure space (Ω,Σ,μ). For a real Banach space $(X, | | \cdot | |_{X})$ let E(X) be a subspace of the space L⁰(X) of μ-equivalence classes of strongly Σ-measurable functions f: Ω → X and consisting of all those f ∈ L⁰(X) for which the scalar function ${| | f (\cdot) | |}_{X}$ belongs to E. Let E(X)˜ stand for the order dual of E(X). For u ∈ E⁺ let $D_{u} (= {f \in E (X) : | | f (\cdot) | |}_{X} \leq u)$ stand for the order interval in E(X). For a real Banach space $(Y, | | \cdot | |_{Y})$ a linear operator T: E(X) → Y is said to be order-bounded whenever for each u ∈...

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Bojan Magajna (1993)

Studia Mathematica

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Ioana Ghenciu (2011)

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Locally pointwise superconvergence of the tensor-product finite element in three dimensions

Jinghong Liu, Liu, Wen, Qiding Zhu (2019)

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Consider a second-order elliptic boundary value problem in three dimensions with locally smooth coefficients and solution. Discuss local superconvergence estimates for the tensor-product finite element approximation on a regular family of rectangular meshes. It will be shown that, by the estimates for the discrete Green’s function and discrete derivative Green’s function, and the relationship of norms in the finite element space such as $L^{2}$ -norms, $W^{1, \infty}$ -norms, and negative-norms in locally...

$^{\infty}$ -vectors and boundedness

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The following two questions as well as their relationship are studied: (i) Is a closed linear operator in a Banach space bounded if its $^{\infty}$ -vectors coincide with analytic (or semianalytic) ones? (ii) When are the domains of two successive powers of the operator in question equal? The affirmative answer to the first question is established in case of paranormal operators. All these investigations are illustrated in the context of weighted shifts.

A note on a class of homeomorphisms between Banach spaces

Piotr Fijałkowski (2005)

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On strongly $l_{p}$ -summing m-linear operators

Lahcène Mezrag (2008)

Colloquium Mathematicae

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We introduce and study a new concept of strongly $l_{p}$ -summing m-linear operators in the category of operator spaces. We give some characterizations of this notion such as the Pietsch domination theorem and we show that an m-linear operator is strongly $l_{p}$ -summing if and only if its adjoint is $l_{p}$ -summing.

Modulation space estimates for multilinear pseudodifferential operators

Árpád Bényi, Kasso A. Okoudjou (2006)

Studia Mathematica

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We prove that for symbols in the modulation spaces $ℳ^{p, q}$ , p ≥ q, the associated multilinear pseudodifferential operators are bounded on products of appropriate modulation spaces. In particular, the symbols we study here are defined without any reference to smoothness, but rather in terms of their time-frequency behavior.