Interpolation of compact operators by general interpolation methods
نویسندگان
چکیده
منابع مشابه
Complex Interpolation of Compact Operators
If (A0, A1) and (B0, B1) are Banach couples and a linear operator T : A0 +A1 → B0 +B1 maps A0 compactly into B0 and maps A1 boundedly into B1, does T necessarily also map [A0, A1]θ compactly into [B0, B1]θ for θ ∈ (0, 1)? After 42 years this question is still not answered, not even in the case where T : A1 → B1 is also compact. But affirmative answers are known for many special choices of (A0, ...
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After 41 years it is still not known whether an operator acting on a Banach pair and which acts compactly on one or both of the “endpoint” spaces also acts compactly on the complex interpolation spaces generated by the pair. We report some recent steps towards solving this and related problems.
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Suppose that (A0, A1) and (B0, B1) are Banach couples, and that T is a linear operator which maps A0 compactly into B0 and A1 boundedly (or even compactly) into B1. Does this imply that T : [A0, A1]θ → [B0, B1]θ compactly? (Here, as usual, [A0, A1]θ denotes the complex interpolation space of Alberto Calderón.) This question has been open for 44 years. Affirmative answers are known for it in man...
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where the infimum is taken over all pairs x^ÇzXj satisfying (2.1). One easily checks that (2.2) gives a norm on X 0 + X i when X0 and X% are compatible. Moreover, when X0+Xi is equipped with this norm, it becomes a Banach space (cf. [3]). Let Cfc denote the set of complex valued functions of the complex variable f = ̂ +irf which are continuous on bounded subsets of S, where fl is the strip 0 <£ ...
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ژورنال
عنوان ژورنال: Journal of Functional Analysis
سال: 2015
ISSN: 0022-1236
DOI: 10.1016/j.jfa.2014.11.017