Summing inclusion maps between symmetric sequence spaces, a survey

Defant, Andreas; Mastyło, Mieczysław; Michels, Carsten

doi:10.1016/s0304-0208(01)80035-9

Cited by 23 publications

(37 citation statements)

References 27 publications

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“…In what follows, we present further results, where we do not need the assumption that the spaces of multipliers between symmetric spaces form the same space. This result can be applied to many concrete symmetric sequence spaces, e.g., Lorentz spaces ℓ p,q and d(w, p), as well as Orlicz spaces ℓ ϕ , via the results on Banach ideal properties of corresponding inclusion maps related to those spaces, presented in [9] and [10]. …”

Section: Interpolation Orbitsmentioning

confidence: 99%

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Interpolation methods of means and orbits

Mastyło

2005

Studia Math.

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Abstract. Banach operator ideal properties of the inclusion maps between Banach sequence spaces are used to study interpolation of orbit spaces. Relationships between those spaces and the method-of-means spaces generated by couples of weighted Banach sequence spaces with the weights determined by concave functions and their Janson sequences are shown. As an application we obtain the description of interpolation orbits in couples of weighted L p -spaces when they are not described by the K-method. We also develop a connection between the method of means with a quasi-parameter and the real method of interpolation generated by the Calderón-Lozanovsky space parameters. Applications to interpolation of operators are also discussed.

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Section: Interpolation Orbitsmentioning

confidence: 99%

“…It is proved in [9] that if E is a p-concave Banach symmetric sequence space with 1 < p ≤ 2, then the inclusion map E ֒→ ℓ p is (M (ℓ p , E), 2)-summing. Thus, by the Banach ideal property, it follows that S :…”

Section: Interpolation Orbitsmentioning

confidence: 99%

Interpolation methods of means and orbits

Mastyło

2005

Studia Math.

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“…Taking 1 < p ≤ 2, we easily conclude by the above argument that the assumptions on concavity and convexity (by duality) in Theorem 4.1 and Clearly, both theorems apply to operators acting on classical complex Banach symmetric sequence spaces including Orlicz sequence spaces ϕ , or Lorentz sequence spaces d(p, w), and in particular p,q . In fact, for these spaces there is a nice description of p-convexity and q-concavity in terms of the parameters generating these spaces (for details see, e.g., [4] and the references therein). To give an example, we present the following statement where part (i) is the analogue of König's result (ii) from the introduction for Lorentz sequence spaces.…”

Section: Theorem 42 Let E and F Be Symmetric Banach Sequence Spacesmentioning

confidence: 99%

Eigenvalue estimates for operators on symmetric Banach sequence spaces

Defant

Mastyło²,

Michels

2003

Proc. Amer. Math. Soc.

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Abstract. Using abstract interpolation theory, we study eigenvalue distribution problems for operators on complex symmetric Banach sequence spaces. More precisely, extending two well-known results due to König on the asymptotic eigenvalue distribution of operators on p-spaces, we prove an eigenvalue estimate for Riesz operators on p-spaces with 1 < p < 2, which take values in a p-concave symmetric Banach sequence space E → p, as well as a dual version, and show that each operator T on a 2-convex symmetric Banach sequence space F , which takes values in a 2-concave symmetric Banach sequence space E, is a Riesz operator with a sequence of eigenvalues that forms a multiplier from F into E. Examples are presented which among others show that the concavity and convexity assumptions are essential.

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“…For a Banach sequence space E containing 2 , the Banach operator ideal of (E, 2)-summing operators consists of all (bounded linear) operators T between Banach spaces for which { T (x n ) } ∈ E for all weakly 2-summable sequences {x n }. Mainly basing on interpolation theory, several key results within the theory of (q, 2)-summing operators and its applications have recently been extended to the more general case of (E, 2)-summing operators (see [4]- [6]). …”

Section: Introductionmentioning

confidence: 99%

“…We refer to [4] and [5] for non-trivial examples of (E, p)-summing opera-tors. We present new simple examples involving symmetric Banach function spaces on finite non-atomic measure spaces (for the theory of these spaces see, e.g., [10] and [12]).…”

Section: Introductionmentioning

confidence: 99%

Composition of (E,2)-summing operators

Defant

Mastyło

2003

Studia Math.

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Abstract. The Banach operator ideal of (q, 2)-summing operators plays a fundamental role within the theory of s-number and eigenvalue distribution of Riesz operators in Banach spaces. A key result in this context is a composition formula for such operators due to H. König, J. R. Retherford and N. Tomczak-Jaegermann. Based on abstract interpolation theory, we prove a variant of this result for (E, 2)-summing operators, E a symmetric Banach sequence space.

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Summing inclusion maps between symmetric sequence spaces, a survey

Cited by 23 publications

References 27 publications

Interpolation methods of means and orbits

Interpolation methods of means and orbits

Eigenvalue estimates for operators on symmetric Banach sequence spaces

Composition of (E,2)-summing operators

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