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Real method of interpolation on subcouples of codimension one

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Astashkin S., Sunehag P.

Studia Mathematica

2008

tom 185

nr 2

151-168

We find necessary and sufficient conditions under which the norms of the interpolation spaces $(N₀,N₁)_{θ,q}$ and $(X₀,X₁)_{θ,q}$ 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 $μ = (μₙ)_{n∈ℤ}$ satisfies the inequalities $μₙ ≤ μ_{n+1} ≤ 2μₙ$ (n ∈ ℤ).

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1 Studia Mathematica

1 Astashkin S.

1 Sunehag P.

1 2008

Wyniki wyszukiwania

Real method of interpolation on subcouples of codimension one