Notes on interpolation by the real method between 𝐶(𝑇,𝐴₀) and 𝐶(𝑇,𝐴₁)

Mastyło, Mieczysław

doi:10.1090/S0002-9939-1988-0934872-3

Notes on interpolation by the real method between $C(T,A_ 0)$ and $C(T,A_ 1)$
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by Mieczysław Mastyło

Proc. Amer. Math. Soc. 102 (1988), 945-948

DOI: https://doi.org/10.1090/S0002-9939-1988-0934872-3

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Abstract:

Let $A$ be a Banach space and let $T$ be a compact Hausdorff space. We denote by $C(T,A)$ the Banach space of all $A$-valued continuous functions defined on $T$ endowed with the supremum norm. We show that if $T$ is infinite and $({A_0},{A_1})$ is a Banach couple with ${A_0}$ continuously embedded in ${A_1}$, then the interpolation space ${(C(T,{A_0}),C(T,{A_1}))_{\varphi ,p}}$ is equal to $C\left ( {T,{{\left ( {{A_0},{A_1}} \right )}_{\varphi ,p}}} \right )$ if and only if ${A_0}$ is closed in ${A_1}$.

References

Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275, DOI 10.1007/978-3-642-66451-9
Jerry Bona and Ridgway Scott, Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces, Duke Math. J. 43 (1976), no. 1, 87–99. MR 393887

Real interpolation functors

23

Fernando Cobos, Some spaces in which martingale difference sequences are unconditional, Bull. Polish Acad. Sci. Math. 34 (1986), no. 11-12, 695–703 (1987) (English, with Russian summary). MR 890615
Michael Cwikel and Jaak Peetre, Abstract $K$ and $J$ spaces, J. Math. Pures Appl. (9) 60 (1981), no. 1, 1–49. MR 616007
Jan Gustavsson, A function parameter in connection with interpolation of Banach spaces, Math. Scand. 42 (1978), no. 2, 289–305. MR 512275, DOI 10.7146/math.scand.a-11754
Svante Janson, Per Nilsson, and Jaak Peetre, Notes on Wolff’s note on interpolation spaces, Proc. London Math. Soc. (3) 48 (1984), no. 2, 283–299. With an appendix by Misha Zafran. MR 729071, DOI 10.1112/plms/s3-48.2.283
Mireille Levy, L’espace d’interpolation réel $(A_{0},\,A_{1}){}_{\theta ,\,p}$ contient $l^{p}$, C. R. Acad. Sci. Paris Sér. A-B 289 (1979), no. 14, A675–A677 (French, with English summary). MR 560331

Notes on non-interpolation spaces

Per Nilsson, Reiteration theorems for real interpolation and approximation spaces, Ann. Mat. Pura Appl. (4) 132 (1982), 291–330 (1983). MR 696048, DOI 10.1007/BF01760986