Real method of interpolation on subcouples of codimension one

S. V. Astashkin, and P. Sunehag. "Real method of interpolation on subcouples of codimension one." Studia Mathematica 185.2 (2008): 151-168. <http://eudml.org/doc/284658>.

@article{S2008,
abstract = {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).},
author = {S. V. Astashkin, P. Sunehag},
journal = {Studia Mathematica},
keywords = {interpolation; real method of interpolation; subcouple; weighted spaces; shift operator; spectrum},
language = {eng},
number = {2},
pages = {151-168},
title = {Real method of interpolation on subcouples of codimension one},
url = {http://eudml.org/doc/284658},
volume = {185},
year = {2008},
}

TY - JOUR
AU - S. V. Astashkin
AU - P. Sunehag
TI - Real method of interpolation on subcouples of codimension one
JO - Studia Mathematica
PY - 2008
VL - 185
IS - 2
SP - 151
EP - 168
AB - 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).
LA - eng
KW - interpolation; real method of interpolation; subcouple; weighted spaces; shift operator; spectrum
UR - http://eudml.org/doc/284658
ER -