Real method of interpolation on subcouples of codimension one

S. V. Astashkin; P. Sunehag

Studia Mathematica, Tome 187 (2008), p. 151-168 / Harvested from The Polish Digital Mathematics Library

Résumé

We find necessary and sufficient conditions under which the norms of the interpolation spaces ${(N ₀, N ₁)}_{θ, q}$ and ${(X ₀, X ₁)}_{θ, q}$ are equivalent on N, where N is the kernel of a nonzero functional ψ ∈ (X₀ ∩ X₁)* and $N_{i}$ is the normed space N with the norm inherited from $X_{i}$ (i = 0,1). Our proof is based on reducing the problem to its partial case studied by Ivanov and Kalton, where ψ is bounded on one of the endpoint spaces. As an application we completely resolve the problem of when the range of the operator $T_{θ} = S - 2^{θ} I$ (S denotes the shift operator and I the identity) is closed in any $ℓ_{p} (μ)$ , where the weight $μ = {(μ ₙ)}_{n \in ℤ}$ satisfies the inequalities $μ ₙ \leq μ_{n + 1} \leq 2 μ ₙ$ (n ∈ ℤ).

Publié le : 2008-01-01

Zbl 1144.46018

EUDML-ID : urn:eudml:doc:284658

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     title = {Real method of interpolation on subcouples of codimension one},
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     year = {2008},
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S. V. Astashkin; P. Sunehag. Real method of interpolation on subcouples of codimension one. Studia Mathematica, Tome 187 (2008) pp. 151-168. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm185-2-4/