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Continuous restrictions of linear maps between Banach spaces. (English) Zbl 0715.47002

Summary: In this note, which emerged from a question of the first-named author at the 17th Winter school on Abstract Analysis, we investigate under what circumstances linear mappings between Banach spaces have continuous restrictions to infinite dimensional subspaces. Two typical cases for which the answer is positive are:
1) If the source space is \(\ell^ 1\) or
2) if the source space and the target space both are Hilbert spaces.
On the other hand, for separable spaces X not containing \(\ell^ 1\) isomorphically, there is a linear map T: \(X\to \ell^ 1\) such that for no infinite dimensional subspace Z of X the restriction \(T| Z\) is continuous. Some additional remarks and questions are also included.

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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